Exercises
B
Ex. 2
$(\rightarrow)$ Trivial.
$(\leftarrow)$ Consider
$$ v_i = \begin{bmatrix} 0 \\\\ \vdots \\\\ 1 \\\\ \vdots \\\\ 0 \end{bmatrix} $$Where only ith position is $1$. Then \begin{align*} (Av_i)[Col \ i] &= (Bv_i)[Col \ i] \quad \text{By hypothesis} \\ &= A[Col \ i] \quad \text{Multiplication by 1} \end{align*} Then it follows $A[Col \\ i] = B[Col \\ i]$ for all $i$.