Coordinate change in a vector space

Question

Suppose $A = \{x_1, x_2\}$ is a basis for some vector space $V$, and that $T: V \rightarrow V$ is a linear transformation. Let $B = \{y_1, y_2\}$ be another basis for $V$ with $y_1 = x_1 + x_2$ and $y_2 = 2x_1 + 3x_2$. If $$T_A = \begin{bmatrix} 3 & 1 \\ -2 & 4\end{bmatrix}$$ I wish to find $T_B$. How would I go about doing this? Thanks.

Answer 1

We only have to "read" the images of the vectors $x_1$ and $x_2$ and then write them in function of $y_1$ and $y_2$.
$T(x_1)=3x_1-2x_2$ and $T(x_2)=x_1+4x_2$. Since the map is linear we can now observe that $$T(y_1)=T(x_1+x_2)=T(x_1)+T(x_2)=4x_1+2x_2$$ $$T(y_2)=T(2x_1+3x_2)=2T(x_1)+3T(x_2)=6x_1-4x_2+3x_1+12x_2=9x_1+8x_2.$$ Now we have $T(y_1)=8y_1-2y_2$ and $T(y_2)=11y_1-y_2$, therefore $$\mathcal M_B^B(T)=T_B=\begin{pmatrix} 8 && 11\\-2 && -1\end{pmatrix}.$$

In order to find $T(y_i)$ I solved the systems $$T(y_1)=\alpha y_1+\beta y_2\text{ and }T(y_2)=\alpha y_1+\beta y_2$$