Linear Algebra, 2020a

16 Chapter One. Linear Systems
above. We like matrix notation because it lightens the clerical load, the copying
of variables and the writing of +’s and =’s.
2.7 Example We can abbreviate this linear system
with this matrix.
x + 2y = 4
y − z = 0
x + 2z = 4
⎛
⎜
1 2 0 4
⎞
⎟
⎝0 1 −1 0⎠
1 0 2 4
The vertical bar reminds a reader of the difference between the coefficients on
the system’s left hand side and the constants on the right. With a bar, this is
an augmented matrix.
⎛
⎜
1 2 0 4
⎞ ⎛
⎟
⎝0 1 −1 0⎠ −ρ 1+ρ 3 ⎜
1 2 0 4
⎞ ⎛
⎟
−→ ⎝0 1 −1 0⎠ 2ρ 2+ρ 3 ⎜
1 2 0 4
⎞
⎟
−→ ⎝0 1 −1 0⎠
1 0 2 4
0 −2 2 0
0 0 0 0
The second row stands for y − z = 0 and the first row stands for x + 2y = 4 so
the solution set is {(4 − 2z, z, z) | z ∈ R}.
Matrix notation also clarifies the descriptions of solution sets. Example 2.3’s
{(2 − 2z + 2w, −1 + z − w, z, w) | z, w ∈ R} is hard to read. We will rewrite it
to group all of the constants together, all of the coefficients of z together, and
all of the coefficients of w together. We write them vertically, in one-column
matrices. ⎛ ⎞ ⎞ ⎞
2 −2 2
⎛
−1
{ ⎜ ⎟
⎝ 0 ⎠ + ⎜
⎝
0
1
1
0
⎛
⎟
⎠ · z + −1
⎜ ⎟ · w | z, w ∈ R}
⎝ 0 ⎠
1
For instance, the top line says that x = 2 − 2z + 2w and the second line says
that y =−1 + z − w. (Our next section gives a geometric interpretation that
will help us picture the solution sets.)
2.8 Definition A column vector, often just called a vector, is a matrix with a
single column. A matrix with a single row is a row vector. The entries of
a vector are sometimes called components. A column or row vector whose
components are all zeros is a zero vector.
Vectors are an exception to the convention of representing matrices with
capital roman letters. We use lower-case roman or greek letters overlined with an