A First Course in Linear Algebra, 2017a

4.11. Orthogonality and the Gram Schmidt Process 229
4.11.1 Orthogonal and Orthonormal Sets
In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal.
First, it is necessary to review some important concepts. You may recall the definitions for the span
of a set of vectors and a linear independent set of vectors. We include the definitions and examples here
for convenience.
Definition 4.116: Span of a Set of Vectors and Subspace
The collection of all linear combinations of a set of vectors {⃗u 1 ,···,⃗u k } in R n is known as the span
of these vectors and is written as span{⃗u 1 ,···,⃗u k }.
We call a collection of the form span{⃗u 1 ,···,⃗u k } a subspace of R n .
Consider the following example.
Example 4.117: Span of Vectors
Describe the span of the vectors ⃗u = [ 1 1 0 ] T and⃗v =
[
3 2 0
] T ∈ R 3 .
Solution. You can see that any linear combination of the vectors ⃗u and ⃗v yields a vector [ x y 0 ] T in
the XY-plane.
Moreover every vector in the XY-plane is in fact such a linear combination of the vectors ⃗u and ⃗v.
That’s because
⎡
⎣
x
y
0
⎤
⎡
⎦ =(−2x + 3y) ⎣
1
1
0
⎤
⎡
⎦ +(x − y) ⎣
3
2
0
⎤
⎦
Thus span{⃗u,⃗v} is precisely the XY-plane.
♠
The span of a set of a vectors in R n is what we call a subspace of R n . A subspace W is characterized
by the feature that any linear combination of vectors of W is again a vector contained in W.
Another important property of sets of vectors is called linear independence.
Definition 4.118: Linearly Independent Set of Vectors
A set of non-zero vectors {⃗u 1 ,···,⃗u k } in R n is said to be linearly independent if no vector in that
set is in the span of the other vectors of that set.
Here is an example.
Example 4.119: Linearly Independent Vectors
Consider vectors ⃗u = [ 1 1 0 ] T ,⃗v =
[
3 2 0
] T ,and⃗w =
[
4 5 0
] T ∈ R 3 .Verifywhether
the set {⃗u,⃗v,⃗w} is linearly independent.