matrix multiplication perspectives

Challenge 2.3 Appendix

Take these two matrices

$A = [142536]$

$B = 135246$

$A \cdot B = C = [22492864]$

from Challenge 2.3.

$A_{1,} = [123]$

$B_{, 1} = 135$

$A_{1,} \cdot B_{, 1} = 1 \cdot 1 + 2 \cdot 3 + 3 \cdot 5 = 22$

$A \cdot B_{, 1} = [142536] \cdot 135 = 1 \cdot [14] + 3 \cdot [25] + 5 \cdot [36] = [2249]$

Columns of $C$ are linear combinations of columns of $A$

$A_{1,} \cdot B = [123] \cdot 135246 = 1 \cdot [12] + 2 \cdot [34] + 3 \cdot [56] = [2228]$

Rows of $C$ are linear combinations of rows of $B$

$A_{, j} \cdot B_{i,} = [14] \cdot [12] + [25] \cdot [34] + [36] \cdot [56] =$

$= [1428] + [615820] + [15301836] = [22492864]$