Hi I need to know how to create a matrix $\displaystyle [T]_{B}$
where $\displaystyle V=P_{2}(\mathbb{R})$ is the vectorspace and
T(f(x))=f(0) +f(1)(x+x^2)
B={1,x,x^2}, the standard basis.
There is a standard technique for constructing the matrix corresponding to a given linear transformation in a given basis:
1) Apply the linear transformation to each basis member in turn.
2) Write the result as a linear combination of the basis members.
3) The coefficients in that linear combination form the columns of the matrix.
For example, the first basis "vector" is the constant function 1: f(x)= 1 so f(0)= 1 and f(1)= 1. T(1)= 1+ 1(x+ x^2= 1+ 1x+ 1x^2. The coefficients are "1 1 1" so the first column of the matrix is $\displaystyle \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}$.
The next basis "vector" is f(x)= x. f(0)= 0 and f(1)= 1. What is T(x)? What is the second column of the matrix?