1. ## Matrix Representation

Let X = (1, 2x - 1) be a basis for R2. Suppose that T is a linear transformation such that T(x) = x + 1/2 and T(1) = 2x + 1. Find RxxT the matrix representation of T with respect to the basis X.

Does this mean-

(1)(A) = T(1)
(2x - 1)(B) = T(x)

and the resulting matrix representation, RxxT is made up of A and B? Thanks.

2. I presume you mean that {1, 2x-1} is a basis for the space of linear polynomials in x. That is NOT usually written as "R2".

But I really don't understand what you mean by
(1)(A) = T(1)
(2x - 1)(B) = T(x)
and the resulting matrix representation, RxxT is made up of A and B
A simple way of finding the matrix representation of a linear transformation in a given basis is to apply the transformation to each basis element in turn, writing the result as a linear combination of vectors in that basis. The coefficients in that linear combination give the columns of the basis.

Here, T(1)= 2x+ 1= a(1)+ b(2x-1)= 2bx+ a- b so we must have 2b= 2 and a- b= 1. That is, b= 1, and then a= 2. The first column of the matrix representation is $\begin{bmatrix}2 \\ 1\end{bmatrix}$.

T(2x- 1)= 2T(x)- T(1)= 2(x+ 1/2)- (2x+ 1)= 2x- 1- 2x+ 1= 0. The second column of the matrix representation is $\begin{bmatrix} 0 \\ 0\end{bmatrix}$.

The matrix representation is $\begin{bmatrix}2 & 0 \\ 1 & 0\end{bmatrix}$.