Linear Algebra

222 Chapter 3. Maps Between Spaces
3.1 Example Multiplying by an appropriately-sized zero matrix from the left
or from the right
� �� �
0 0 1 3 2
=
0 0 −1 1 −1
results in a zero matrix.
� 0 0 0
0 0 0
� � �� �
2 3 0 0
=
1 4 0 0
� �
0 0
0 0
After zero matrices, the matrices whose actions are easiest to understand
are the ones with a single nonzero entry.
3.2 Definition A matrix with all zeroes except for a one in the i, j entry is an
i, j unit matrix.
3.3 Example This is the 1, 2 unit matrix with three rows and two columns,
multiplying from the left.
⎛ ⎞ ⎛ ⎞
0 1 � � 7 8
⎝0
0⎠
5 6
= ⎝0
0⎠
7 8
0 0
0 0
Acting from the left, an i, j unit matrix copies row j of the multiplicand into
row i of the result. From the right an i, j unit matrix copies column i of the
multiplicand into column j of the result.
⎛
1
⎝4
2
5
⎞ ⎛
3 0
6⎠⎝0
⎞ ⎛
1 0
0⎠
= ⎝0
⎞
1
4⎠
7 8 9 0 0 0 7
3.4 Example Rescaling these matrices simply rescales the result. This is the
action from the left of the matrix that is twice the one in the prior example.
⎛ ⎞ ⎛ ⎞
0 2 � � 14 16
⎝0
0⎠
5 6
= ⎝ 0 0⎠
7 8
0 0
0 0
And this is the action of the matrix that is minus three times the one from the
prior example.
⎛
1
⎝4 2
5
⎞ ⎛
3 0
6⎠⎝0
⎞ ⎛
−3 0
0 ⎠ = ⎝0 ⎞
−3
−12⎠
7 8 9 0 0 0 −21
Next in complication are matrices with two nonzero entries. There are two
cases. If a left-multiplier has entries in different rows then their actions don’t
interact.