Linear Algebra, 2020a

40 Chapter One. Linear Systems
linear system and parametrize: x = 2 − y/2 − z/2.
⎛ ⎞
2
⎛ ⎞
−1/2
⎛ ⎞
−1/2
P = { ⎝0⎠ + y · ⎝
0
1
0
⎠ + z · ⎝ 0
1
⎠ | y, z ∈ R}
Shown in grey are the vectors associated with y and z, offset from the origin
by 2 units along the x-axis, so that their entire body lies in the plane. Thus the
vector sum of the two, shown in black, has its entire body in the plane along
with the rest of the parallelogram.
Generalizing, a set of the form {⃗p + t 1 ⃗v 1 + t 2 ⃗v 2 + ···+ t k ⃗v k | t 1 ,...,t k ∈ R}
where ⃗v 1 ,...,⃗v k ∈ R n and k n is a k-dimensional linear surface (or k-flat).
For example, in R 4 ⎞ ⎛ ⎞
2 1
π
{⎛
⎜ ⎟
⎝ 3 ⎠ + t 0
⎜ ⎟
⎝0⎠ | t ∈ R}
−0.5 0
is a line,
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 1 2
0
{ ⎜ ⎟
⎝0⎠ + t 1
⎜ ⎟
⎝ 0 ⎠ + s 0
⎜ ⎟ | t, s ∈ R}
⎝1⎠ 0 −1 0
is a plane, and
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
3 0 1 2
1
{ ⎜ ⎟
⎝−2⎠ + r 0
⎟ ⎟⎟⎠ 0
⎜ + s ⎜ ⎟
⎝ 0 ⎝1⎠ + t 0
⎜ ⎟ | r, s, t ∈ R}
⎝1⎠ 0.5 −1 0 0
is a three-dimensional linear surface. Again, the intuition is that a line permits
motion in one direction, a plane permits motion in combinations of two directions,
etc. When the dimension of the linear surface is one less than the dimension of
the space, that is, when in R n we have an (n − 1)-flat, the surface is called a
hyperplane.
A description of a linear surface can be misleading about the dimension. For
example, this
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 2
0
L = { ⎜ ⎟
⎝−1⎠ + t 1
⎜ ⎟
⎝ 0 ⎠ + s 2
⎜ ⎟ | t, s ∈ R}
⎝ 0 ⎠
−2 −1 −2