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3. Find the matrix representation for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D^{2}+2D+1_{P_{3}}:P_{3}\to P_{3}}
with respect to the basis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1, t, t^2, t^3}
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| Solution:
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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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = D^2 + 2D + 1_{P_3}}
then:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(1)=0+2\cdot 0+1=1={\begin{bmatrix}1\\0\\0\\0\end{bmatrix}}}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(t)=0+2\cdot 1+t=2+t={\begin{bmatrix}2\\1\\0\\0\end{bmatrix}}}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \leftarrow }
Fixed error here
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{bmatrix}1&2&2&0\\0&1&4&6\\0&0&1&6\\0&0&0&1\end{bmatrix}}}
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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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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}}}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle E_{21}={\begin{bmatrix}0&0\\1&0\end{bmatrix}}}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle E_{12}={\begin{bmatrix}0&1\\0&0\end{bmatrix}}}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle E_{22}={\begin{bmatrix}0&0\\0&1\end{bmatrix}}}
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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}}}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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}}}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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}}}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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  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}}}
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: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{bmatrix}1&2&2&0\\0&1&4&6\\0&0&1&6\\0&0&0&1\end{bmatrix}}}
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7. Compute a matrix representation for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L:{\text{Mat}}_{2\times 2}(\mathbb {F} )\to {\text{Mat}}_{1\times 2}(\mathbb {F} )}
defined by:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(X)={\begin{bmatrix}1&-1\end{bmatrix}}X}
using the standard bases.
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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_{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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