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
$L(t^{2})=2+2\cdot 2t+t^{2}=2+4t+t^{2}{\begin{bmatrix}2\\4\\1\\0\end{bmatrix}}$
$L(t^{3})=6t+2\cdot 3t^{2}+t^{3}=6t+6t^{2}+t^{3}={\begin{bmatrix}0\\6\\6\\1\end{bmatrix}}$
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 ${\begin{bmatrix}a&c\\b&d\end{bmatrix}}$ and consider the map $R_{A}:{\text{Mat}}_{2\times 2}(\mathbb {F} )\to {\text{Mat}}_{2\times 2}(\mathbb {F} )$ defined by $R_{A}(X)=XA$. Compute the matrix representation of this linear map with respect to the basis:

$E_{11}={\begin{bmatrix}1&0\\0&0\end{bmatrix}}$

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

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

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

Solution As before we evaluate the function on the basis elements and represent the outputs as coordinate vectors.
$R_{A}(E_{11})=E_{11}A={\begin{bmatrix}1&0\\0&0\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}a&c\\0&0\end{bmatrix}}={\begin{bmatrix}a\\0\\c\\0\end{bmatrix}}$
$R_{A}(E_{21})=E_{21}A={\begin{bmatrix}0&0\\1&0\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}0&0\\a&c\end{bmatrix}}={\begin{bmatrix}0\\a\\0\\c\end{bmatrix}}$
$R_{A}(E_{12})=E_{12}A={\begin{bmatrix}0&1\\0&0\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}b&d\\0&0\end{bmatrix}}={\begin{bmatrix}b\\0\\d\\0\end{bmatrix}}$
$R_{A}(E_{22})=E_{22}A={\begin{bmatrix}0&0\\0&1\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}0&0\\b&d\end{bmatrix}}={\begin{bmatrix}0\\b\\0\\d\end{bmatrix}}$
This gives the matrix representation of $R_{A}$ as ${\begin{bmatrix}a&0&b&0\\0&a&0&b\\c&0&d&0\\0&c&0&d\end{bmatrix}}$ $L(t^{3})=6t+2\cdot 3t^{2}+t^{3}=6t+6t^{2}+t^{3}={\begin{bmatrix}0\\6\\6\\1\end{bmatrix}}$
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:
$L(E_{11})={\begin{bmatrix}1&-1\end{bmatrix}}E_{11}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}1&0\\0&0\end{bmatrix}}={\begin{bmatrix}1&0\end{bmatrix}}={\begin{bmatrix}1\\0\end{bmatrix}}$
$L(E_{12})={\begin{bmatrix}1&-1\end{bmatrix}}E_{12}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}0&1\\0&0\end{bmatrix}}={\begin{bmatrix}0&1\end{bmatrix}}={\begin{bmatrix}0\\1\end{bmatrix}}$
$L(E_{21})={\begin{bmatrix}1&-1\end{bmatrix}}E_{21}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}0&0\\1&0\end{bmatrix}}={\begin{bmatrix}-1&0\end{bmatrix}}={\begin{bmatrix}-1\\0\end{bmatrix}}$
$L(E_{22})={\begin{bmatrix}1&-1\end{bmatrix}}E_{22}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}0&0\\0&1\end{bmatrix}}={\begin{bmatrix}0&-1\end{bmatrix}}={\begin{bmatrix}0\\-1\end{bmatrix}}$
This gives the matrix representation: ${\begin{bmatrix}1&0&-1&0\\0&1&0&-1\end{bmatrix}}$