I'm trying the solve the following problem:
Verify, whether
matrices $A=\begin{pmatrix} -1&1 \\ 1&2 \end{pmatrix}$,
$B=\begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix}$, $C= \begin{pmatrix}
1&1 \\ 0&0 \end{pmatrix}$ and $D=\begin{pmatrix} 0&0 \\ 1&2
\end{pmatrix}$ form a basis for $\mathbb{R}_{2\times 2}$. Also
find coordinates of matrix $E=\begin{pmatrix} -2&2 \\ 8&4
\end{pmatrix}$ in the basis.
What I did was to create a matrix $F=$ $\begin{pmatrix}
-1&0 &1 &0 \\
1&1 &0 &1 \\
1&1 &1 &0 \\
2&0 &0 &2
\end{pmatrix}$ and verify, whether there are any linearly dependent lines. I found out that there are none, therefore the matrices $A,B,C,D$ are generating the entire space and therefore they form a basis. Is my solution correct?
And how would I find the coordinates of the matrix E in the basis? Do I just put the numbers from the matrix $E$ as solutions in my matrix $F$? Thanks!
Best Answer
You found they form a basis. Now you need to solve $aA+bB+cC+dD=E$ for $a,b,c$ and $d$. This is a linear system of $4$ equations in $4$ unknowns. You could form another matrix to solve.
Here's the matrix: $\left(\begin {array}{rrrr|r}-1&0&1&0&-2\\1&1&1&0&2\\1&1&0&1&8\\2&0&0&2&4\end{array}\right)$.