Problem 4. (10 points)
If
are arbitrary vectors in $\mathbb{R}^{2 \times 2}$, then the mapping
$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$ and $B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$
$(A, B) = a_{11}b_{11} + a_{12}b_{12} + a_{21}b_{21} + a_{22}b_{22}$
defines an inner product in $\mathbb{R}^{2 \times 2}$. Use this inner product to determine $(A, B)$, $||A||$, $||B||$, and the angle $\alpha_{A, B}$ between $A$ and $B$ for
$(A, B) = \boxed{}
||A|| = \boxed{}
||B|| = \boxed{}
$\alpha_{A, B} = \boxed{}$ (radians).
Note: You can earn partial credit on this problem.
A = $\begin{bmatrix} -5 & -4 \\ -1 & 4 \end{bmatrix}$ and B = $\begin{bmatrix} -3 & 3 \\ 2 & 1 \end{bmatrix}$
