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ubuntu2004
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<?xml version='1.0' encoding='UTF-8'?>

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<exercise xmlns="https://spatext.clontz.org" version="0.0">

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  <statement>

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      <p>Consider the following statement.</p>

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        <ul><li> The set of

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            {{#context_pol}}

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                polynomials <m>{{set}}</m>

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                {{#task_ind}}is linearly independent.{{/task_ind}}

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                {{#task_spa}}spans <m>\mathcal{P}_3</m>.{{/task_spa}}

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            {{/context_pol}}

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            {{#context_mat}}

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                matrices <m>{{set}}</m>

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                {{#task_ind}}is linearly independent.{{/task_ind}}

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                {{#task_spa}}spans <m>\mathrm{M}_{2,2}</m>.{{/task_spa}}

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            {{/context_mat}}

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        </li></ul>

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    <ol>

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        <li> Write an equivalent statement using a {{#context_pol}}polynomial{{/context_pol}}{{#context_mat}}matrix{{/context_mat}} equation.</li>

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        <li> Explain why your statement is true or false.</li>

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    </ol>

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  </statement>

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  <answer>

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      <p><me>\operatorname{RREF}{{matrix}}={{rref}}</me></p>

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     <ol>

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         <li>

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             {{#task_ind}}

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                 The {{#context_pol}}polynomial{{/context_pol}}{{#context_mat}}matrix{{/context_mat}} equation

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                 <me>{{equation}}</me>

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                 has no nontrivial solutions.

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             {{/task_ind}}

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             {{#task_spa}}

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                 {{#context_pol}}

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                     The polynomial equation <me>{{equation}}=f</me> has a solution for every <m>f \in \mathcal{P}_3</m>.

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                 {{/context_pol}}

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                 {{#context_mat}}

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                     The matrix equation <me>{{equation}}=B</me> has a solution for every <m>B \in \mathrm{M}_{2,2}</m>.

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                 {{/context_mat}}

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             {{/task_spa}}

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          </li>

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          <li>

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              The set of {{#context_pol}}polynomials{{/context_pol}}{{#context_mat}}matrices{{/context_mat}} <m>{{set}}</m>

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             {{#task_ind}}

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                 is linearly {{#result}}in{{/result}}dependent.

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             {{/task_ind}}

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             {{#task_spa}}

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                 {{#result}}

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                     spans

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                 {{/result}}

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                 {{^result}}

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                     does not span

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                 {{/result}}

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                 {{#context_pol}}

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                     <m>\mathcal{P}_3</m>.

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                 {{/context_pol}}

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                 {{#context_mat}}

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                     <m>\mathrm{M}_{2,2}</m>.

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                 {{/context_mat}}

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             {{/task_spa}}

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          </li>

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      </ol>

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  </answer>

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</exercise>

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