<item ident="V1-0003" title="V1 | Vector spaces | ver. 0003">
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<div class="exercise-statement">
<p>
<strong>V1.</strong>
</p>
<p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"/> be the set of all pairs <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(x,y)" alt="(x,y)" title="(x,y)" data-latex="(x,y)"/> of real numbers together with the following operations: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 5\right)" alt="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 5\right)" title="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 5\right)" data-latex="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 5\right)"/>
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?c \odot (x,y) = \left(c x,\,c y\right) ." alt="c \odot (x,y) = \left(c x,\,c y\right) ." title="c \odot (x,y) = \left(c x,\,c y\right) ." data-latex="c \odot (x,y) = \left(c x,\,c y\right) ."/>
</p>
<p> (a) Show that vector addition is associative, that is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)." alt="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)." title="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)." data-latex="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)."/>
</p>
<p> (b) Explain why <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"/> nonetheless is not a vector space. </p>
</div>
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<mattext texttype="text/html"><div class="exercise-statement">
<p>
<strong>V1.</strong>
</p>
<p> Let <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"> be the set of all pairs <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(x,y)" alt="(x,y)" title="(x,y)" data-latex="(x,y)"> of real numbers together with the following operations: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?(x_1,y_1)%5Coplus%20(x_2,y_2)=%20%5Cleft(x_%7B1%7D%20+%20x_%7B2%7D,%5C,y_%7B1%7D%20+%20y_%7B2%7D%20-%205%5Cright)" alt="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 5\right)" title="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 5\right)" data-latex="(x_1,y_1)\oplus (x_2,y_2)= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 5\right)">
</p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?c%20%5Codot%20(x,y)%20=%20%5Cleft(c%20x,%5C,c%20y%5Cright)%20." alt="c \odot (x,y) = \left(c x,\,c y\right) ." title="c \odot (x,y) = \left(c x,\,c y\right) ." data-latex="c \odot (x,y) = \left(c x,\,c y\right) .">
</p>
<p> (a) Show that vector addition is associative, that is: </p>
<p style="text-align:center;">
<img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cleft((x_1,y_1)%5Coplus(x_2,y_2)%5Cright)%5Coplus(x_3,y_3)=(x_1,y_1)%5Coplus%5Cleft((x_2,y_2)%5Coplus(x_3,y_3)%5Cright)." alt="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)." title="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right)." data-latex="\left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right).">
</p>
<p> (b) Explain why <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"> nonetheless is not a vector space. </p>
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<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"/> is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold: </p>
<ul>
<li>scalar multiplication does not distribute over vector addition</li>
<li>scalar multiplication does not distribute over scalar addition</li>
</ul>
</div>
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<mattext texttype="text/html"><div class="exercise-answer">
<h4>Partial Answer:</h4>
<p><img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?V" alt="V" title="V" data-latex="V"> is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold: </p>
<ul>
<li>scalar multiplication does not distribute over vector addition</li>
<li>scalar multiplication does not distribute over scalar addition</li>
</ul>
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