Linear Algebra

16 Chapter 1. Linear Systems
Scalar multiplication can be written in either order: r · �v or �v · r, or without
the ‘·’ symbol: r�v. (Do not refer to scalar multiplication as ‘scalar product’
because that name is used for a different operation.)
2.12 Example
⎛
⎝ 2
⎞ ⎛
3⎠
+ ⎝
1
3
⎞ ⎛
−1⎠
= ⎝
4
2+3
⎞ ⎛
3 − 1⎠
= ⎝
1+4
5
⎛ ⎞
⎞ 1
⎜
2⎠
7 · ⎜ 4 ⎟
⎝−1⎠
5
−3
=
⎛ ⎞
7
⎜ 28 ⎟
⎝ −7 ⎠
−21
Notice that the definitions of vector addition and scalar multiplication agree
where they overlap, for instance, �v + �v =2�v.
With the notation defined, we can now solve systems in the way that we will
use throughout this book.
2.13 Example This system
2x + y − w =4
y + w + u =4
x − z +2w =0
reduces in this way.
⎛
2
⎝0
1
1
0
0
−1
1
0
1
⎞
4
4⎠
1 0 −1 2 0 0
−(1/2)ρ1+ρ3
−→
(1/2)ρ2+ρ3
−→
⎛
2
⎝0
1
1
0
0
−1
1
0
1
⎞
4
4 ⎠
0
⎛
2
⎝0
−1/2
1 0
1 0
−1 5/2
−1 0
1 1
0 −2
⎞
4
4⎠
0 0 −1 3 1/20 The solution set is {(w +(1/2)u, 4 − w − u, 3w +(1/2)u, w, u) � � w, u ∈ R}. We
write that in vector form.
⎛ ⎞
x
⎜
⎜y
⎟
{ ⎜
⎜z
⎟
⎝w⎠
u
=
⎛ ⎞
0
⎜
⎜4
⎟
⎜
⎜0
⎟
⎝0⎠
0
+
⎛ ⎞ ⎛ ⎞
1 1/2
⎜
⎜−1⎟
⎜
⎟ ⎜−1
⎟
⎜ 3 ⎟ w + ⎜
⎜1/2
⎟
⎝ 1 ⎠ ⎝ 0 ⎠
0 1
u � � w, u ∈ R}
Note again how well vector notation sets off the coefficients of each parameter.
For instance, the third row of the vector form shows plainly that if u is held
fixed then z increases three times as fast as w.
That format also shows plainly that there are infinitely many solutions. For
example, we can fix u as 0, let w range over the real numbers, and consider the
first component x. We get infinitely many first components and hence infinitely
many solutions.