Section 1.7 Homework

3. Find the matrix representation for $D^{2}+2D+1_{P_{3}}:P_{3}\to P_{3}$ with respect to the basis $1,t,t^{2},t^{3}$.

Solution:

In order to calculate the matrix representation, we evaluate the function on each of the basis elements and then write the coordinate vector for the output of the function in terms of the same basis. In particular if we let $L=D^{2}+2D+1_{P_{3}}$ then: $L(1)=0+2\cdot 0+1=1={\begin{bmatrix}1\\0\\0\\0\end{bmatrix}}$ $L(t)=0+2\cdot 1+t=2+t={\begin{bmatrix}2\\1\\0\\0\end{bmatrix}}$ $\leftarrow$ Fixed error here $L(t^{2})=2+2\cdot 2t+t^{2}=2+4t+t^{2}{\begin{bmatrix}2\\4\\1\\0\end{bmatrix}}$ Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(t^{3})=6t+2\cdot 3t^{2}+t^{3}=6t+6t^{2}+t^{3}={\begin{bmatrix}0\\6\\6\\1\end{bmatrix}}} Which gives the matrix representation: ${\begin{bmatrix}1&2&2&0\\0&1&4&6\\0&0&1&6\\0&0&0&1\end{bmatrix}}$

6. Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{bmatrix}a&c\\b&d\end{bmatrix}}} and consider the map $R_{A}:{\text{Mat}}_{2\times 2}(\mathbb {F} )\to {\text{Mat}}_{2\times 2}(\mathbb {F} )$ defined by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R_{A}(X)=XA} . Compute the matrix representation of this linear map with respect to the basis:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle E_{11}={\begin{bmatrix}1&0\\0&0\end{bmatrix}}}

$E_{21}={\begin{bmatrix}0&0\\1&0\end{bmatrix}}$

$E_{12}={\begin{bmatrix}0&1\\0&0\end{bmatrix}}$

$E_{22}={\begin{bmatrix}0&0\\0&1\end{bmatrix}}$

Solution:

As before we evaluate the function on the basis elements and represent the outputs as coordinate vectors. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R_{A}(E_{11})=E_{11}A={\begin{bmatrix}1&0\\0&0\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}a&c\\0&0\end{bmatrix}}={\begin{bmatrix}a\\0\\c\\0\end{bmatrix}}} $R_{A}(E_{21})=E_{21}A={\begin{bmatrix}0&0\\1&0\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}0&0\\a&c\end{bmatrix}}={\begin{bmatrix}0\\a\\0\\c\end{bmatrix}}$ $R_{A}(E_{12})=E_{12}A={\begin{bmatrix}0&1\\0&0\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}b&d\\0&0\end{bmatrix}}={\begin{bmatrix}b\\0\\d\\0\end{bmatrix}}$ $R_{A}(E_{22})=E_{22}A={\begin{bmatrix}0&0\\0&1\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}0&0\\b&d\end{bmatrix}}={\begin{bmatrix}0\\b\\0\\d\end{bmatrix}}$ This gives the matrix representation of $R_{A}$ as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{bmatrix}a&0&b&0\\0&a&0&b\\c&0&d&0\\0&c&0&d\end{bmatrix}}} $L(t^{3})=6t+2\cdot 3t^{2}+t^{3}=6t+6t^{2}+t^{3}={\begin{bmatrix}0\\6\\6\\1\end{bmatrix}}$ Which gives the matrix representation: ${\begin{bmatrix}1&2&2&0\\0&1&4&6\\0&0&1&6\\0&0&0&1\end{bmatrix}}$

7. Compute a matrix representation for $L:{\text{Mat}}_{2\times 2}(\mathbb {F} )\to {\text{Mat}}_{1\times 2}(\mathbb {F} )$ defined by:
$L(X)={\begin{bmatrix}1&-1\end{bmatrix}}X$ using the standard bases.

Solution:

We again calculate: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(E_{11})={\begin{bmatrix}1&-1\end{bmatrix}}E_{11}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}1&0\\0&0\end{bmatrix}}={\begin{bmatrix}1&0\end{bmatrix}}={\begin{bmatrix}1\\0\end{bmatrix}}} Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(E_{12})={\begin{bmatrix}1&-1\end{bmatrix}}E_{12}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}0&1\\0&0\end{bmatrix}}={\begin{bmatrix}0&1\end{bmatrix}}={\begin{bmatrix}0\\1\end{bmatrix}}} Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(E_{21})={\begin{bmatrix}1&-1\end{bmatrix}}E_{21}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}0&0\\1&0\end{bmatrix}}={\begin{bmatrix}-1&0\end{bmatrix}}={\begin{bmatrix}-1\\0\end{bmatrix}}} Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(E_{22})={\begin{bmatrix}1&-1\end{bmatrix}}E_{22}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}0&0\\0&1\end{bmatrix}}={\begin{bmatrix}0&-1\end{bmatrix}}={\begin{bmatrix}0\\-1\end{bmatrix}}} This gives the matrix representation: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{bmatrix}1&0&-1&0\\0&1&0&-1\end{bmatrix}}}

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