# Finding a matrix from Linear Transformations

I'm having trouble understanding the steps in working with bases. Suppose T is a linear transformation from $$R^2$$ to $$R^2$$ defined by $$T(x,y)$$ = $$(2x-y, x+y)$$ and we want to find the matrix of T with respect to the bases $$B_1 = [(1,1),(2,1)] \ \ \text{ and }B_2 = [(-1,2), (1,1)].$$

I know the matrix $$A[u_1]B_1 = [T(u_1)B_2$$ but i'm not sure how to apply that with the information given.

• In general do you know how to represent a linear transformation using a matrix (given a basis)? Jun 8, 2020 at 4:18
• It might be useful to note that matrix $B_{1}$ when written as follows $\begin{bmatrix}1 & 2 \\ 1 & 1 \end{bmatrix}$ (i.e. basis vectors in the columns) indicates that if you right multiply it by $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ you obtain $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and likewise, if you right multiply it by $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ you obtain $\begin{bmatrix}2 \\ 1\end{bmatrix}$. I hope this helps. Jun 8, 2020 at 23:17
• I know that B and T can form matrices, but I think the formula mentioned is the only one I know, so I was confused as to the value of $u$ Jun 10, 2020 at 3:39

Let us denote the standard basis $$e_1 := \begin{pmatrix} 1 \\ 0 \end{pmatrix}\quad\textrm{and}\quad e_2 := \begin{pmatrix} 0 \\ 1 \end{pmatrix}.$$ In this basis $$T = \begin{pmatrix} 2 & -1 \\ 1 & 1 \end{pmatrix}.$$ The basis transformation matrices from $$(e_1,e_2)$$ to $$B_1$$ or $$B_2$$ are given by $$B_1 = \begin{pmatrix} 1 & 1 \\ 2 & 1 \end{pmatrix}\quad\textrm{and}\quad B_2 = \begin{pmatrix} -1 & 2 \\ 1 & 1 \end{pmatrix}.$$ The transformation matrix in each basis will be given by $$TB_1^{-1} \quad\textrm{and}\quad TB_2^{-1}.$$
• Ah so there is no $[u]$ required. But if $T$ had a vector, say $u = (2,3)$ then it would have to be multiplied to that formula? Jun 10, 2020 at 3:36