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I am given two bases of a vector space consisting of matrices.

e basis:

e1=`\begin{bmatrix}1&2\\0&5\end{bmatrix} e2=\begin{bmatrix}1&1\\-1&0\end{bmatrix} e3= \begin{bmatrix}1&0\\2&3\end{bmatrix} e4=\begin{bmatrix}1&2\\4&3\end{bmatrix}

The second basis is the e^ basis:

e^1=\begin{bmatrix}-2&2\\-3&3\end{bmatrix} e^2=\begin{bmatrix}0&1\\2&3\end{bmatrix} e^3=\begin{bmatrix}1&1\\-1&0\end{bmatrix} e^4=\begin{bmatrix}-1&2\\5&2\end{bmatrix} Compute the transition matrix P from e to e^.

I know I have to write each e of the original basis as a linear combination of the e^ matrices, but how do I actually compute that?

Matthew Orlić
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1 Answers1

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$$\begin{bmatrix}1&2\\0&5\end{bmatrix}= c_1\begin{bmatrix}-2&2\\-3&3\end{bmatrix}+c_2\begin{bmatrix}0&1\\2&3\end{bmatrix}+c_3\begin{bmatrix}1&1\\-1&0\end{bmatrix}+c_4\begin{bmatrix}-1&2\\5&2\end{bmatrix}$$

Solve for $c_1,c_2,c_3,c_4$ to find the coordinates of the $$\begin{bmatrix}1&2\\0&5\end{bmatrix}$$ with respect to the second basis.

Similarly you find coordinates of the other $3$ basis elements.

The transition matrix is a matrix whose columns are these $4$ coordinate vectors.

Mohammad Riazi-Kermani
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