Suppose that vectors v1 = (1, 2), v2 = (2,−1), and that the basis B is B=v1, v2 . (this is a list)
Let T be the linear transformation from R2 to R2 given by T(v1) = v1 and T(v2) = 0.
(a) Write down the matrix for T in the new basis B. (You should be able to do this
directly from the definition of T).
(b) Use this to write down the matrix for T in the standard basis.

Question

Suppose that vectors v1 = (1, 2), v2 = (2,−1), and that the basis B is B=v1, v2 . (this is a list)
Let T be the linear transformation from R2 to R2 given by T(v1) = v1 and T(v2) = 0.
(a) Write down the matrix for T in the new basis B. (You should be able to do this
directly from the definition of T).
(b) Use this to write down the matrix for T in the standard basis.

amistre64 · Answer

$$\vec{x}=\binom{x_1}{x_2};\ B=\{v_1,v_2\}$$
$$[\vec{x}]_B=x_1v_1+x_2v_2$$
$$\begin{align} T([\vec{x}]_B)&=T(x_1v_1+x_2v_2)\\
&=T(x_1v_1)+T(x_2v_2)\\
&=x_1T(v_1)+x_2T(v_2)\
\end{align}$$
$$T_B=\{T(v_1),T(v_2)\}$$
$$T_e=\{T(e_1),T(e_2)\}$$
and that is the best that I can come up with :)