How to find the matrix of Linear Transformation with respect to S given the linear transformations of the matrix?

I was trying to attempt the following question: $Let, L: $P1 -> P1 be defined by:

$L(t+1) = t - 1, L(t-1) = 2t + 1 $

A) Find the matrix of L with respect to the basis $S = {t+1, t-1}$ for P1.

B) Find $L(2t+3)$ using definition of L and Also using the matrix obtained in (A)

Now I tried to approach the solution by using the V1 V2 | L(V1) | (L(V1) to find the matrix

Then I tried the (B) Part but the value by matrix and definition don't match

The solution is appended below as a reference, Sorry couldn't write all of that using the Text hence a picture is appended:

Any help will be much appreciated

Solution 1:

Your matrix is correct. You also start correctly by representing $2t+3$ in the basis: $$2t+3 = 2.5v_1 - 0.5v_2.$$ Hence by linearity $$L(2t+3) = L(2.5v_1 - 0.5v_2) = 2.5L(v_1) - 0.5L(v_2) = 2.5(t-1)-0.5(2t+1) = 1.5t-3.$$ This should be your final answer.

Lets try to get the same result using the matrix: $$\begin{pmatrix} 0 & 1.5\\\ 1 & 0.5\end{pmatrix}\begin{pmatrix} 2.5\\ -0.5 \end{pmatrix} = \begin{pmatrix} -0.75\\ 2.25 \end{pmatrix}.$$ This is in our special basis $\{t+1,t-1\}.$ Let's bring this back to the normal basis $\{1,t\}:$ $$-0.75(t+1) + 2.25(t-1) = 1.5t-3.$$