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<h2 class="hd hd-2 unit-title">Pset 6 checker</h2>
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<div class="vert vert-0" data-id="block-v1:MITx+ES.1803+2023_Fall+type@html+block@7cf3f49a63c841bd8b0ce471961b753d">
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<p><a href="/assets/courseware/v1/9532f2409edc9c82f70a13e1085c8222/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/mites_1803_s24_ps6.pdf" target="[object Object]">Problem set 6 (PDF)</a></p>
<p></p>
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data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-1a-d-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-1a-d-check/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="ps6-1a-d-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-1a-d-check-problem-progress" tabindex="-1">
Problem 1a-d
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-1a-d-check-problem-progress"></div>
<div class="problem">
<div>
<p>
To check your answers put them in the appropriate box and click the &#8216;Check' button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. </p>
<p>
Unless otherwise stated the checker expects numerical answers to 3 significant digits. </p>
<p>
For formulas you should use Matlab notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] <b class="bf">NOT </b> [mathjaxinline]3x[/mathjaxinline]. </p>
<p>
The checker knows many of the standard functions. Use exp(x) and ln(x) for the exponential and natural log functions. </p>
<p>
The formula checker expects a specific letter for the variables, so be sure to use the one specified. </p>
<p>
For this problem enter columns or rows as a list inside square brackets, e.g. [1,2,3] or [[mathjaxinline]x^2+y[/mathjaxinline], [mathjaxinline]x*y[/mathjaxinline]]. Enter matrices in the following form [[x,y,z],[1,2,3]]. That is each row is within brackets within the outer brackets. So, [x,y,z] is the first row and [1,2,3] is the second row. </p>
<p>
<b class="bf">1(a)</b>
<p style="display:inline">[mathjaxinline]A\mathbf{b} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps6-1a-d-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-1a-d-check_2_1" id="input_ps6-1a-d-check_2_1" aria-describedby="status_ps6-1a-d-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-1a-d-check_2_1"/>
<span class="status unanswered" id="status_ps6-1a-d-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-1a-d-check_2_1" class="answer"/>
<div id="display_ps6-1a-d-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-1a-d-check_2_1_dynamath" name="input_ps6-1a-d-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<b class="bf">1(b)</b>
<b class="bf">No checker given: Matlab should have given an error message. Report that message on your pset.</b>
</p>
<p>
<b class="bf">1(c)</b>
<p style="display:inline">[mathjaxinline]\det (A) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps6-1a-d-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-1a-d-check_3_1" id="input_ps6-1a-d-check_3_1" aria-describedby="status_ps6-1a-d-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-1a-d-check_3_1"/>
<span class="status unanswered" id="status_ps6-1a-d-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-1a-d-check_3_1" class="answer"/>
<div id="display_ps6-1a-d-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-1a-d-check_3_1_dynamath" name="input_ps6-1a-d-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">1(d)</b> Give [mathjaxinline]A^{-1}[/mathjaxinline]. Entries should be correct to 3 decimal places. </p>
<p>
<p style="display:inline">[mathjaxinline]A^{-1} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps6-1a-d-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
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<div class="unanswered inline">
<input type="text" name="input_ps6-1a-d-check_4_1" id="input_ps6-1a-d-check_4_1" aria-describedby="status_ps6-1a-d-check_4_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-1a-d-check_4_1"/>
<span class="status unanswered" id="status_ps6-1a-d-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-1a-d-check_4_1" class="answer"/>
<div id="display_ps6-1a-d-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-1a-d-check_4_1_dynamath" name="input_ps6-1a-d-check_4_1_dynamath"/>
</div>
</div></div>
</p>
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<input type="hidden" name="problem_id" value="Problem 1a-d" />
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<h3 class="hd hd-3 problem-header" id="ps6-1e-g-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-1e-g-check-problem-progress" tabindex="-1">
Problem 1e-g
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-1e-g-check-problem-progress"></div>
<div class="problem">
<div>
<p>
For this problem vectors and matrices should be input in the same form as in problem 1a-d. </p>
<p><b class="bf">1(e)</b> Give [mathjaxinline]\mathbf{x}[/mathjaxinline]. Entries should be correct to 3 decimal places. </p>
<p>
<p style="display:inline">[mathjaxinline]\mathbf{x} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps6-1e-g-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-1e-g-check_2_1" id="input_ps6-1e-g-check_2_1" aria-describedby="status_ps6-1e-g-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-1e-g-check_2_1"/>
<span class="status unanswered" id="status_ps6-1e-g-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-1e-g-check_2_1" class="answer"/>
<div id="display_ps6-1e-g-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-1e-g-check_2_1_dynamath" name="input_ps6-1e-g-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">1(f)</b> Give the RREF of [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline]. </p>
<p>
<p style="display:inline">RREF(A) = </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps6-1e-g-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-1e-g-check_3_1" id="input_ps6-1e-g-check_3_1" aria-describedby="status_ps6-1e-g-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-1e-g-check_3_1"/>
<span class="status unanswered" id="status_ps6-1e-g-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-1e-g-check_3_1" class="answer"/>
<div id="display_ps6-1e-g-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-1e-g-check_3_1_dynamath" name="input_ps6-1e-g-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">RREF(B) = </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps6-1e-g-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-1e-g-check_4_1" id="input_ps6-1e-g-check_4_1" aria-describedby="status_ps6-1e-g-check_4_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-1e-g-check_4_1"/>
<span class="status unanswered" id="status_ps6-1e-g-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-1e-g-check_4_1" class="answer"/>
<div id="display_ps6-1e-g-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-1e-g-check_4_1_dynamath" name="input_ps6-1e-g-check_4_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">1(g)</b> [mathjaxinline]A^ T[/mathjaxinline] and [mathjaxinline]B^ T[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]A^ T =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ps6-1e-g-check_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-1e-g-check_5_1" id="input_ps6-1e-g-check_5_1" aria-describedby="status_ps6-1e-g-check_5_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-1e-g-check_5_1"/>
<span class="status unanswered" id="status_ps6-1e-g-check_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-1e-g-check_5_1" class="answer"/>
<div id="display_ps6-1e-g-check_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-1e-g-check_5_1_dynamath" name="input_ps6-1e-g-check_5_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]B^ T =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_ps6-1e-g-check_6_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-1e-g-check_6_1" id="input_ps6-1e-g-check_6_1" aria-describedby="status_ps6-1e-g-check_6_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-1e-g-check_6_1"/>
<span class="status unanswered" id="status_ps6-1e-g-check_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-1e-g-check_6_1" class="answer"/>
<div id="display_ps6-1e-g-check_6_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-1e-g-check_6_1_dynamath" name="input_ps6-1e-g-check_6_1_dynamath"/>
</div>
</div></div>
</p>
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</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-2-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-graded="False" data-block-type="problem" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-2-check" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-request-token="44833908383111f0ad1b0e09798aea1b">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps6-2-check" class="problems-wrapper" role="group"
aria-labelledby="ps6-2-check-problem-title"
data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-2-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-2-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps6-2-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-2-check-problem-progress" tabindex="-1">
Problem 2
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-2-check-problem-progress"></div>
<div class="problem">
<div>
<p>
<b class="bf">2(a)</b>
<p style="display:inline">RREF(A) = </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps6-2-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-2-check_2_1" id="input_ps6-2-check_2_1" aria-describedby="status_ps6-2-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-2-check_2_1"/>
<span class="status unanswered" id="status_ps6-2-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2-check_2_1" class="answer"/>
<div id="display_ps6-2-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-2-check_2_1_dynamath" name="input_ps6-2-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<b class="bf">2(b)</b>
<p style="display:inline">Rank of [mathjaxinline]A[/mathjaxinline] = </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps6-2-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-2-check_3_1" id="input_ps6-2-check_3_1" aria-describedby="status_ps6-2-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-2-check_3_1"/>
<span class="status unanswered" id="status_ps6-2-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2-check_3_1" class="answer"/>
<div id="display_ps6-2-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-2-check_3_1_dynamath" name="input_ps6-2-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">2(c)</b> Give the basis of the column space of [mathjaxinline]A[/mathjaxinline]. Enter the vectors in the three boxes below. If the basis has fewer than 3 vectors you can leave boxes empty. </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div class="multi-inputs-group" role="group"><p style="display:inline">Vector 1: <div id="inputtype_ps6-2-check_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ps6-2-check_4_1" id="input_ps6-2-check_4_1" aria-describedby="status_ps6-2-check_4_1" value=""/>
<span class="trailing_text" id="trailing_text_ps6-2-check_4_1"/>
<span class="status unanswered" id="status_ps6-2-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2-check_4_1" class="answer"/>
</div>
</div></p>
<br/>
<p style="display:inline">Vector 2: <div id="inputtype_ps6-2-check_4_2" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ps6-2-check_4_2" id="input_ps6-2-check_4_2" aria-describedby="status_ps6-2-check_4_2" value=""/>
<span class="trailing_text" id="trailing_text_ps6-2-check_4_2"/>
<span class="status unanswered" id="status_ps6-2-check_4_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2-check_4_2" class="answer"/>
</div>
</div></p>
<br/>
<p style="display:inline">Vector 3: <div id="inputtype_ps6-2-check_4_3" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ps6-2-check_4_3" id="input_ps6-2-check_4_3" aria-describedby="status_ps6-2-check_4_3" value=""/>
<span class="trailing_text" id="trailing_text_ps6-2-check_4_3"/>
<span class="status unanswered" id="status_ps6-2-check_4_3" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2-check_4_3" class="answer"/>
</div>
</div></p>
</div></div>
</p>
<p><b class="bf">2(d)</b> Give the basis of the null space of [mathjaxinline]A[/mathjaxinline]. Enter the vectors in the three boxes below. If the null space is trivial you can enter an empty set of brackets, i.e. [], in one of the boxes. </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div class="multi-inputs-group" role="group"><p style="display:inline">Vector 1: <div id="inputtype_ps6-2-check_5_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ps6-2-check_5_1" id="input_ps6-2-check_5_1" aria-describedby="status_ps6-2-check_5_1" value=""/>
<span class="trailing_text" id="trailing_text_ps6-2-check_5_1"/>
<span class="status unanswered" id="status_ps6-2-check_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2-check_5_1" class="answer"/>
</div>
</div></p>
<br/>
<p style="display:inline">Vector 2: <div id="inputtype_ps6-2-check_5_2" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ps6-2-check_5_2" id="input_ps6-2-check_5_2" aria-describedby="status_ps6-2-check_5_2" value=""/>
<span class="trailing_text" id="trailing_text_ps6-2-check_5_2"/>
<span class="status unanswered" id="status_ps6-2-check_5_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2-check_5_2" class="answer"/>
</div>
</div></p>
<br/>
<p style="display:inline">Vector 3: <div id="inputtype_ps6-2-check_5_3" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ps6-2-check_5_3" id="input_ps6-2-check_5_3" aria-describedby="status_ps6-2-check_5_3" value=""/>
<span class="trailing_text" id="trailing_text_ps6-2-check_5_3"/>
<span class="status unanswered" id="status_ps6-2-check_5_3" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2-check_5_3" class="answer"/>
</div>
</div></p>
</div></div>
</p>
<p>
<b class="bf">2(e)</b>
<em>No answer checker given.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 2" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps6-2-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps6-2-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-2-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-2-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-2-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-3-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-graded="False" data-block-type="problem" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-3-check" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-request-token="44833908383111f0ad1b0e09798aea1b">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps6-3-check" class="problems-wrapper" role="group"
aria-labelledby="ps6-3-check-problem-title"
data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-3-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-3-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps6-3-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-3-check-problem-progress" tabindex="-1">
Problem 3
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-3-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">3.</b> Again, there are an infinite number of possible solutions. The checker only knows about one. Though it is probably the one most people will arrive at. </p>
<p>
Give the basis of the space of homogeneous solutions a comma separated list of functions. Use the variable t. </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps6-3-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-3-check_2_1" id="input_ps6-3-check_2_1" aria-describedby="status_ps6-3-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-3-check_2_1"/>
<span class="status unanswered" id="status_ps6-3-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-3-check_2_1" class="answer"/>
<div id="display_ps6-3-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-3-check_2_1_dynamath" name="input_ps6-3-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
Give the constant solution: </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps6-3-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-3-check_3_1" id="input_ps6-3-check_3_1" aria-describedby="status_ps6-3-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-3-check_3_1"/>
<span class="status unanswered" id="status_ps6-3-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-3-check_3_1" class="answer"/>
<div id="display_ps6-3-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-3-check_3_1_dynamath" name="input_ps6-3-check_3_1_dynamath"/>
</div>
</div></div>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 3" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps6-3-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps6-3-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-3-check-problem-title">
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<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-3-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-3-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-5" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-4-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-graded="False" data-block-type="problem" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-4-check" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-request-token="44833908383111f0ad1b0e09798aea1b">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
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<div id="problem_ps6-4-check" class="problems-wrapper" role="group"
aria-labelledby="ps6-4-check-problem-title"
data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-4-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-4-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps6-4-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-4-check-problem-progress" tabindex="-1">
Problem 4
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-4-check-problem-progress"></div>
<div class="problem">
<div>
<b class="bf">4a-d.</b>
<em>No answer checker given.</em>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 4" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps6-4-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps6-4-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-4-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-4-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-4-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-6" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-5-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-graded="False" data-block-type="problem" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-5-check" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-request-token="44833908383111f0ad1b0e09798aea1b">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps6-5-check" class="problems-wrapper" role="group"
aria-labelledby="ps6-5-check-problem-title"
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data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
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<h3 class="hd hd-3 problem-header" id="ps6-5-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-5-check-problem-progress" tabindex="-1">
Problem 5
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-5-check-problem-progress"></div>
<div class="problem">
<div>
<p>
<b class="bf">5(a)</b>
<em>No answer checker given.</em>
</p>
<p><b class="bf">5(b)</b> Give the functions [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. These should be in amplitude phase form. </p>
<p><b class="bf">Warning:</b> The checker will accept answers in rectangular form, but this is not what the pset asks for. </p>
<p>
<p style="display:inline">[mathjaxinline]x(t) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps6-5-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-5-check_2_1" id="input_ps6-5-check_2_1" aria-describedby="status_ps6-5-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-5-check_2_1"/>
<span class="status unanswered" id="status_ps6-5-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-5-check_2_1" class="answer"/>
<div id="display_ps6-5-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-5-check_2_1_dynamath" name="input_ps6-5-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]y(t) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps6-5-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-5-check_3_1" id="input_ps6-5-check_3_1" aria-describedby="status_ps6-5-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-5-check_3_1"/>
<span class="status unanswered" id="status_ps6-5-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-5-check_3_1" class="answer"/>
<div id="display_ps6-5-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-5-check_3_1_dynamath" name="input_ps6-5-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<b class="bf">5(c)</b>
<em>No answer checker given.</em>
</p>
<p>
<b class="bf">5(d)</b>
<em>No answer checker given.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 5" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps6-5-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps6-5-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-5-check-problem-title">
</span>
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<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-5-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-5-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-7" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-6-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-graded="False" data-block-type="problem" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-6-check" data-init="XBlockToXModuleShim" data-runtime-class="LmsRuntime" data-request-token="44833908383111f0ad1b0e09798aea1b">
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{"xmodule-type": "Problem"}
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data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-6-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-6-check/handler/xmodule_handler"
data-problem-score="0"
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<h3 class="hd hd-3 problem-header" id="ps6-6-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-6-check-problem-progress" tabindex="-1">
Problem 6
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-6-check-problem-progress"></div>
<div class="problem">
<div>
<p>
<b class="bf">6(a)</b>
</p>
<p>
<p style="display:inline">Rank([mathjaxinline]A[/mathjaxinline]) = </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps6-6-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-6-check_2_1" id="input_ps6-6-check_2_1" aria-describedby="status_ps6-6-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-6-check_2_1"/>
<span class="status unanswered" id="status_ps6-6-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-6-check_2_1" class="answer"/>
<div id="display_ps6-6-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-6-check_2_1_dynamath" name="input_ps6-6-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<b class="bf">6(b)</b>
</p>
<p>
<p style="display:inline">RREF([mathjaxinline]A[/mathjaxinline]) = </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps6-6-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-6-check_3_1" id="input_ps6-6-check_3_1" aria-describedby="status_ps6-6-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-6-check_3_1"/>
<span class="status unanswered" id="status_ps6-6-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-6-check_3_1" class="answer"/>
<div id="display_ps6-6-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-6-check_3_1_dynamath" name="input_ps6-6-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<b class="bf">6(c)</b>
<em>No answer checker given.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 6" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps6-6-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps6-6-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-6-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-6-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-6-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-8" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-7-check">
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<div id="problem_ps6-7-check" class="problems-wrapper" role="group"
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data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-7-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-7-check/handler/xmodule_handler"
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Problem 7
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps6-7-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">7(a)</b> Give 3 eigenvalue/vector pairs. Your answer should include the constant k. </p>
<p><b class="bf">Warning:</b> The checker is not that clever. It will mark it correct if you give the same pair twice. The graders will undoubtedly mark this wrong. You should have three distinct pairs. Also, the checker will mark both eigenvalue and eigenvector wrong if just one of them is wrong. </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div class="multi-inputs-group" role="group"><p style="display:inline">Eigenvalue: <div id="inputtype_ps6-7-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-7-check_2_1" id="input_ps6-7-check_2_1" aria-describedby="status_ps6-7-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-7-check_2_1"/>
<span class="status unanswered" id="status_ps6-7-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-7-check_2_1" class="answer"/>
<div id="display_ps6-7-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-7-check_2_1_dynamath" name="input_ps6-7-check_2_1_dynamath"/>
</div>
</div></p>
<br/>
<p style="display:inline">Eigenvector: <div id="inputtype_ps6-7-check_2_2" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-7-check_2_2" id="input_ps6-7-check_2_2" aria-describedby="status_ps6-7-check_2_2" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-7-check_2_2"/>
<span class="status unanswered" id="status_ps6-7-check_2_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-7-check_2_2" class="answer"/>
<div id="display_ps6-7-check_2_2" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-7-check_2_2_dynamath" name="input_ps6-7-check_2_2_dynamath"/>
</div>
</div></p>
</div></div>
</p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div class="multi-inputs-group" role="group"><p style="display:inline">Eigenvalue: <div id="inputtype_ps6-7-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-7-check_3_1" id="input_ps6-7-check_3_1" aria-describedby="status_ps6-7-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-7-check_3_1"/>
<span class="status unanswered" id="status_ps6-7-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-7-check_3_1" class="answer"/>
<div id="display_ps6-7-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-7-check_3_1_dynamath" name="input_ps6-7-check_3_1_dynamath"/>
</div>
</div></p>
<br/>
<p style="display:inline">Eigenvector: <div id="inputtype_ps6-7-check_3_2" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-7-check_3_2" id="input_ps6-7-check_3_2" aria-describedby="status_ps6-7-check_3_2" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-7-check_3_2"/>
<span class="status unanswered" id="status_ps6-7-check_3_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-7-check_3_2" class="answer"/>
<div id="display_ps6-7-check_3_2" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-7-check_3_2_dynamath" name="input_ps6-7-check_3_2_dynamath"/>
</div>
</div></p>
</div></div>
</p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div class="multi-inputs-group" role="group"><p style="display:inline">Eigenvalue: <div id="inputtype_ps6-7-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-7-check_4_1" id="input_ps6-7-check_4_1" aria-describedby="status_ps6-7-check_4_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-7-check_4_1"/>
<span class="status unanswered" id="status_ps6-7-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-7-check_4_1" class="answer"/>
<div id="display_ps6-7-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-7-check_4_1_dynamath" name="input_ps6-7-check_4_1_dynamath"/>
</div>
</div></p>
<br/>
<p style="display:inline">Eigenvector: <div id="inputtype_ps6-7-check_4_2" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps6-7-check_4_2" id="input_ps6-7-check_4_2" aria-describedby="status_ps6-7-check_4_2" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-7-check_4_2"/>
<span class="status unanswered" id="status_ps6-7-check_4_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-7-check_4_2" class="answer"/>
<div id="display_ps6-7-check_4_2" class="equation">`{::}`</div>
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Problem 8
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There are multiple ways to organize the companion matrix. The answer checker wants the rows in the order [mathjaxinline]x[/mathjaxinline], [mathjaxinline]y[/mathjaxinline], [mathjaxinline]x'[/mathjaxinline], [mathjaxinline]y'[/mathjaxinline]. </p>
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Problem EC-1
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<p><b class="bf">EC-1(a)</b> Give the coefficient matrix for this system. Use the variables V_1, V_2, r. </p>
<p>
<p style="display:inline">Coefficient matrix = </p>
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<p><b class="bf">EC-1(b)</b> To simplify entry, let [mathjaxinline]k_ j = r/V_ j[/mathjaxinline]. Give the coefficient matrix for this system using the variables k_1, k_2, k_3. </p>
<p>
<p style="display:inline">Coefficient matrix = </p>
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<p>
<b class="bf">EC-1(c)</b>
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<p>
<b class="bf">EC-1(d)</b>
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Problem EC-2
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<b class="bf">EC-2(a)</b>
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<p><b class="bf">EC-2(b)</b> Give the [mathjaxinline]4\times 4[/mathjaxinline] companion matrix. </p>
<p>
<p style="display:inline">Companion matrix = </p>
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<p><b class="bf">EC-2(c)</b> Give the eigenvalues of the companion matrix and periods for the normal modes. Each should be a comma separated list of values. Use i for complex [mathjaxinline]i[/mathjaxinline]. </p>
<p>
<p style="display:inline">Eigenvalues = </p>
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<p style="display:inline">Periods = </p>
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<p>
<b class="bf">EC-2(d)</b>
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<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-EC-2-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-EC-2-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps6-EC-2-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>