I've been studying linear algebra and have gotten the general understanding that when we are looking at a vector described by two separate bases, that we can generally apply:
$[u]_{T} = P_{S \rightarrow T}[u]_S$
Now, I am trying to apply this to this problem:
Now, to address this problem I modeled $P_{s \rightarrow T}$ as \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}
Then, I set up my equations, fulfilling the inital equation, but hopefully giving me a system of equations to solve for $a, b, c, d$:
$\begin{bmatrix} a & b \\ c & d \\\end{bmatrix} *\begin{bmatrix} 11 \\ 5 \\\end{bmatrix} = \begin{bmatrix} 4 \\ 3 \\\end{bmatrix}$
$\begin{bmatrix} a & b \\ c & d \\\end{bmatrix} *\begin{bmatrix} 0 \\ -7 \\\end{bmatrix} = \begin{bmatrix} 1 \\ -2 \\\end{bmatrix}$
The answer is shown to be
However, from my system I can see that these are not the values that I would obtain, as from performing a bit of mental matrix multiplication, I can see that $b = -\frac{1}{7}$, and this is obviously wrong. This leads me to believe there is something wrong with my method.
I understand this is very simple math for many people in this community, but would really appreciate someone taking the time to point me in the right direction
Thank You,
David
You need to write your equations the other way around: $$\begin{bmatrix} a & b \\ c & d \\\end{bmatrix} \begin{bmatrix} 4 \\ 3 \\\end{bmatrix} = \begin{bmatrix} 11 \\ 5 \\\end{bmatrix} $$ and $$ \begin{bmatrix} a & b \\ c & d \\\end{bmatrix} \begin{bmatrix} 1 \\ -2 \\\end{bmatrix}=\begin{bmatrix} 0 \\ -7 \\\end{bmatrix} . $$
For matrices:
When you express a linear operator as a matrix, there are two bases involved: one for the domain and one for the codomain. So if you are changing both bases, the change of basis is $$[u]_T=P^{-1}[u]_SP.$$