linear-guest

4.2 Hyperplanes 87
unless any of the vectors v j lives in the (k − 1)-dimensional hyperplane determined
by the other k − 1 vectors
{
0 +
}
k∑
λ i v i | λ i ∈ R .
i≠j
Example 47 (3+1 vectors that do not specify a 3-dimensional hyperplane)
⎧⎛
⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎫
3 1 0 1
⎪⎨
1
0
1
1
S :=
4
⎜1
+ s
0
⎟ ⎜0
+ t
0
⎟ ⎜0
+ u
0
⎪⎬
⎟ ⎜0
s, t, u ∈ R
⎟
⎝5⎠
⎝0⎠
⎝0⎠
⎝0⎠
⎪⎩
9 0 0 0 ⏐
⎪⎭
is not a 3-dimensional hyperplane because
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎧ ⎛ ⎞ ⎛ ⎞
⎫
1 1 0 1 0
1
0
1
0
⎜0
= 1
0
⎟ ⎜0
+ 1
0
⎪⎨
0
1
⎟ ⎜0
∈ s
0
⎟ ⎜0
+ t
0
⎪⎬
⎟ ⎜0
s, t ∈ R .
⎟
⎝0⎠
⎝0⎠
⎝0⎠
⎝0⎠
⎝0⎠
⎪⎩
0 0 0 0 0 ⏐
⎪⎭
In fact, the set could be rewritten as
⎧⎛
⎞ ⎛ ⎞ ⎛ ⎞
⎫
3
1
0
⎪⎨
1
0
1
S =
4
⎜1
+ (s + u)
0
⎟ ⎜0
+ (t + u)
0
⎪⎬
⎟ ⎜0
s, t, u ∈ R
⎟
⎝5⎠
⎝0⎠
⎝0⎠
⎪⎩
9
0
0 ⏐
⎪⎭
⎧⎛
⎞ ⎛ ⎞ ⎛ ⎞
⎫
3 1 0
⎪⎨
1
0
1
=
4
⎜1
+ a
0
⎟ ⎜0
+ b
0
⎪⎬
⎟ ⎜0
a, b ∈ R
⎟
⎝5⎠
⎝0⎠
⎝0⎠
⎪⎩
9 0 0 ⏐
⎪⎭
and so is actually the same 2-dimensional hyperplane in R 6 as in example 46.
87