# computing a matrix given a specific basis.

The question is as follows.

Let $\alpha = \left\{ \begin{bmatrix}1&0\\0&0 \end{bmatrix},\begin{bmatrix}0&1\\0&0 \end{bmatrix},\begin{bmatrix}0&0\\1&0 \end{bmatrix},\begin{bmatrix}0&0\\0&1 \end{bmatrix} \right\}$.

If $A =\begin{bmatrix} 1 & -2 \\ 0 & 4 \end{bmatrix}$, compute $[A]_\alpha$.

How do I do this? The other problems in this section define a linear transformation and ask you to find the matrix with respect to a given basis. Here is a matrix and we must find another matrix? The answer is $(1,-2,0,4)^T$, but I'm not sure why exactly this is the case.

• $[A]_\alpha$ is the coordinate vector of $A$ with respect to the basis $\alpha$. So how can you make $A$ as a linear combination of the basis vectors? You take $1$ of the first one, $-2$ of the second, $0$ of the third, and $4$ of the fourth, which translates to the coordinate vector $(1,-2,0,4)$. – Dave Jun 8 '17 at 1:43
• If your 4 matrices are named $M_1,M_2,M_3,M_4$, it means $A=(1)M_1+(-2)M_2+(0)M_3+(4)M_4$, that's all. – Jean Marie Jun 8 '17 at 1:43

This is intuitive. They are clearly looking for a linear combination of your basis matrices that equal the matrix $A$. Clearly this is
$$A =(1)\begin{bmatrix}1&0\\0&0 \end{bmatrix} -2 \begin{bmatrix}0&1\\0&0 \end{bmatrix} + (0)\begin{bmatrix}0&0\\1&0 \end{bmatrix} + (4)\begin{bmatrix}0&0\\0&1 \end{bmatrix} .$$