Linear Algebra, 2020a

246 Chapter Three. Maps Between Spaces
Next in complication are matrices with two nonzero entries.
3.5 Example There are two cases. If a left-multiplier has entries in different rows
then their actions don’t interact.
⎛
⎜
1 0 0
⎞ ⎛
⎟ ⎜
1 2 3
⎞ ⎛
⎟ ⎜
1 0 0
⎞ ⎛
⎟ ⎜
0 0 0
⎞ ⎛ ⎞
1 2 3
⎟ ⎜ ⎟
⎝0 0 2⎠
⎝4 5 6⎠ =( ⎝0 0 0⎠ + ⎝0 0 2⎠)
⎝4 5 6⎠
0 0 0 7 8 9 0 0 0 0 0 0 7 8 9
⎛
⎜
1 2 3
⎞ ⎛
⎞
0 0 0
⎟ ⎜
⎟
= ⎝0 0 0⎠ + ⎝14 16 18⎠
0 0 0 0 0 0
⎛
⎞
1 2 3
⎜
⎟
= ⎝14 16 18⎠
0 0 0
But if the left-multiplier’s nonzero entries are in the same row then that row of
the result is a combination.
⎛
⎜
1 0 2
⎞ ⎛
⎟ ⎜
1 2 3
⎞ ⎛
⎟ ⎜
1 0 0
⎞ ⎛
⎟ ⎜
0 0 2
⎞ ⎛ ⎞
1 2 3
⎟ ⎜ ⎟
⎝0 0 0⎠
⎝4 5 6⎠ =( ⎝0 0 0⎠ + ⎝0 0 0⎠)
⎝4 5 6⎠
0 0 0 7 8 9 0 0 0 0 0 0 7 8 9
⎛
⎜
1 2 3
⎞ ⎛
⎞
14 16 18
⎟ ⎜
⎟
= ⎝0 0 0⎠ + ⎝ 0 0 0⎠
0 0 0 0 0 0
⎛
⎞
15 18 21
⎜
⎟
= ⎝ 0 0 0⎠
0 0 0
Right-multiplication acts in the same way, but with columns.
3.6 Example Consider the columns of the product of two 2×2 matrices.
( )( ) (
)
g 1,1 g 1,2 h 1,1 h 1,2 g 1,1 h 1,1 + g 1,2 h 2,1 g 1,1 h 1,2 + g 1,2 h 2,2
=
g 2,1 g 2,2 h 2,1 h 2,2 g 2,1 h 1,1 + g 2,2 h 2,1 g 2,1 h 1,2 + g 2,2 h 2,2
Each column is the result of multiplying G by the corresponding column of H.
( ) (
) ( ) (
)
h 1,1 g 1,1 h 1,1 + g 1,2 h 2,1 h 1,2 g 1,1 h 1,2 + g 1,2 h 2,2
G =
G =
h 2,1 g 2,1 h 1,1 + g 2,2 h 2,1 h 2,2 g 2,1 h 1,2 + g 2,2 h 2,2
3.7 Lemma In a product of two matrices G and H, the columns of GH are formed
by taking G times the columns of H
⎛
⎞ ⎛
.
.
⎞
G · ⎜
⎝
⃗h 1 ··· ⃗h n
⎟
⎠ = .
.
⎜
⎝G · ⃗h 1 ··· G · ⃗h n
⎟
⎠
.
.
.
.