Chapter 03 - Section 02

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 $i$.