Exercise 5 — Linear Algebra

Consider this matrix, representing a transformation of $\mathbb{R}^{2}$ with respect to the bases.

\frac{1}{2}\cdot\begin{pmatrix}1&1\\ -1&1\end{pmatrix}\qquad B=\langle\begin{pmatrix}0\\ 1\end{pmatrix},\begin{pmatrix}1\\ 0\end{pmatrix}\rangle\quad D=\langle\begin{pmatrix}1\\ 1\end{pmatrix},\begin{pmatrix}1\\ -1\end{pmatrix}\rangle

To what vector in the codomain is the first member of $B$ mapped?
The second member?
Where is a general vector from the domain (a vector with components $x$ and $y$ ) mapped? That is, what transformation of $\mathbb{R}^{2}$ is represented with respect to $B,D$ by this matrix?

AI Tutor