Linear Algebra, 2020a

18 Chapter One. Linear Systems
We write scalar multiplication either as r · ⃗v or ⃗v · r, and sometimes even
omit the ‘·’ symbol: r⃗v. (Do not refer to scalar multiplication as ‘scalar product’
because that name is for a different operation.)
2.12 Example
⎛ ⎞ ⎛ ⎞
⎛
⎜
2
⎞ ⎛ ⎞ ⎛
3
⎟ ⎜ ⎟ ⎜
2 + 3
⎞ ⎛
⎟ ⎜
5
⎞ 1 7
⎟
4
⎝3⎠ + ⎝−1⎠ = ⎝3 − 1⎠ = ⎝2⎠ 7 · ⎜ ⎟
⎝−1⎠ = 28
⎜ ⎟
⎝ −7 ⎠
1 4 1 + 4 5
−3 −21
Observe that the definitions of addition and scalar multiplication agree where
they overlap; for instance, ⃗v + ⃗v = 2⃗v.
With these definitions, we are set to use matrix and vector notation to both
solve systems and express the solution.
2.13 Example This system
2x + y − w = 4
y + w + u = 4
x − z + 2w = 0
reduces in this way.
⎛
⎜
2 1 0 −1 0 4
⎞
⎟
⎝0 1 0 1 1 4
1 0 −1 2 0 0
−→
⎠ −(1/2)ρ 1+ρ 3
(1/2)ρ 2 +ρ 3
−→
⎛
⎜
2 1 0 −1 0 4
⎞
⎟
⎝0 1 0 1 1 4⎠
0 −1/2 −1 5/2 0 −2
⎛
⎜
2 1 0 −1 0 4
⎞
⎟
⎝0 1 0 1 1 4⎠
0 0 −1 3 1/2 0
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 0 1 1/2
⎜y
⎟ ⎜4
⎟ ⎜−1
⎟ ⎜−1
⎟
{
z
=
0
+
⎜ ⎟ ⎜ ⎟ ⎜
⎝w⎠
⎝0⎠
⎝
u 0
3
1
0
w +
1/2
u | w, u ∈ R}
⎟ ⎜ ⎟
⎠ ⎝ 0 ⎠
1
Note 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 fixed then z
increases three times as fast as w. Another thing shown plainly is that setting