I need help with a linear algebra question that I have been stuck on for some time:
Find the coordinate vectors of $\begin{bmatrix}3\\1\end{bmatrix}$ with respect to the basis {$\begin{bmatrix}1\\1\end{bmatrix}$, $\begin{bmatrix}1\\3\end{bmatrix}$}
I am not so sure of how to begin, however, I think I need to obtain a linear system from this information. Then, I should row-reduce and find a solution. I am really not so sure, though. Can someone please help me?
edit: what I did so far:
I made the matrix
$\begin{bmatrix}1&1&3\\1&3&1\end{bmatrix}$ and row reduced it to find $x_1 = 4$ and $x_2 = -1$.
Does this mean that $x_B$ = $\begin{bmatrix}4\\-1\end{bmatrix}$?
Best Answer
From the comment above by @amd.
That’s absolutely right. The coordinates of the vector relative to the basis are just the coefficients in the linear combination $a(1,1)^T+b(1,3)^T$ that equals $(3,1)^T$, which leads to the system of linear equations in $a$ and $b$ represented by the matrix you constructed.