# Section 1.7

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

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 ${\displaystyle L=D^{2}+2D+1_{P_{3}}}$ then:

${\displaystyle L(1)=0+2\cdot 0+1=1={\begin{bmatrix}1\\0\\0\\0\end{bmatrix}}}$
${\displaystyle L(t)=0+2\cdot 1+t=2+t={\begin{bmatrix}2\\1\\0\\0\end{bmatrix}}}$ ${\displaystyle \leftarrow }$ Fixed error here
${\displaystyle L(t^{2})=2+2\cdot 2t+t^{2}=2+4t+t^{2}{\begin{bmatrix}2\\4\\1\\0\end{bmatrix}}}$
${\displaystyle L(t^{3})=6t+2\cdot 3t^{2}+t^{3}=6t+6t^{2}+t^{3}={\begin{bmatrix}0\\6\\6\\1\end{bmatrix}}}$
Which gives the matrix representation: ${\displaystyle {\begin{bmatrix}1&2&2&0\\0&1&4&6\\0&0&1&6\\0&0&0&1\end{bmatrix}}}$

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

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

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

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

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

Solution:
As before we evaluate the function on the basis elements and represent the outputs as coordinate vectors.

${\displaystyle R_{A}(E_{11})=E_{11}A={\begin{bmatrix}1&0\\0&0\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}a&c\\0&0\end{bmatrix}}={\begin{bmatrix}a\\0\\c\\0\end{bmatrix}}}$
${\displaystyle R_{A}(E_{21})=E_{21}A={\begin{bmatrix}0&0\\1&0\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}0&0\\a&c\end{bmatrix}}={\begin{bmatrix}0\\a\\0\\c\end{bmatrix}}}$
${\displaystyle R_{A}(E_{12})=E_{12}A={\begin{bmatrix}0&1\\0&0\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}b&d\\0&0\end{bmatrix}}={\begin{bmatrix}b\\0\\d\\0\end{bmatrix}}}$
${\displaystyle R_{A}(E_{22})=E_{22}A={\begin{bmatrix}0&0\\0&1\end{bmatrix}}{\begin{bmatrix}a&c\\b&d\end{bmatrix}}={\begin{bmatrix}0&0\\b&d\end{bmatrix}}={\begin{bmatrix}0\\b\\0\\d\end{bmatrix}}}$
This gives the matrix representation of ${\displaystyle R_{A}}$ as ${\displaystyle {\begin{bmatrix}a&0&b&0\\0&a&0&b\\c&0&d&0\\0&c&0&d\end{bmatrix}}}$ ${\displaystyle L(t^{3})=6t+2\cdot 3t^{2}+t^{3}=6t+6t^{2}+t^{3}={\begin{bmatrix}0\\6\\6\\1\end{bmatrix}}}$
Which gives the matrix representation: ${\displaystyle {\begin{bmatrix}1&2&2&0\\0&1&4&6\\0&0&1&6\\0&0&0&1\end{bmatrix}}}$

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

Solution:
We again calculate:

${\displaystyle L(E_{11})={\begin{bmatrix}1&-1\end{bmatrix}}E_{11}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}1&0\\0&0\end{bmatrix}}={\begin{bmatrix}1&0\end{bmatrix}}={\begin{bmatrix}1\\0\end{bmatrix}}}$
${\displaystyle L(E_{12})={\begin{bmatrix}1&-1\end{bmatrix}}E_{12}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}0&1\\0&0\end{bmatrix}}={\begin{bmatrix}0&1\end{bmatrix}}={\begin{bmatrix}0\\1\end{bmatrix}}}$
${\displaystyle L(E_{21})={\begin{bmatrix}1&-1\end{bmatrix}}E_{21}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}0&0\\1&0\end{bmatrix}}={\begin{bmatrix}-1&0\end{bmatrix}}={\begin{bmatrix}-1\\0\end{bmatrix}}}$
${\displaystyle L(E_{22})={\begin{bmatrix}1&-1\end{bmatrix}}E_{22}={\begin{bmatrix}1&-1\end{bmatrix}}{\begin{bmatrix}0&0\\0&1\end{bmatrix}}={\begin{bmatrix}0&-1\end{bmatrix}}={\begin{bmatrix}0\\-1\end{bmatrix}}}$
This gives the matrix representation: ${\displaystyle {\begin{bmatrix}1&0&-1&0\\0&1&0&-1\end{bmatrix}}}$

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