A First Course in Linear Algebra, 2017a

4.11. Orthogonality and the Gram Schmidt Process 243
The orthogonal complement is defined as the set of all vectors which are orthogonal to all vectors in
the original subspace. It turns out that it is sufficient that the vectors in the orthogonal complement be
orthogonal to a spanning set of the original space.
Proposition 4.142: Orthogonal to Spanning Set
Let W be a subspace of R n such that W = span{⃗w 1 ,⃗w 2 ,···,⃗w m }.ThenW ⊥ is the set of all vectors
which are orthogonal to each ⃗w i in the spanning set.
The following proposition demonstrates that the orthogonal complement of a subspace is itself a subspace.
Proposition 4.143: The Orthogonal Complement
Let W be a subspace of R n . Then the orthogonal complement W ⊥ is also a subspace of R n .
Consider the following proposition.
Proposition 4.144: Orthogonal Complement of R n
The complement of R n is the set containing the zero vector:
{ }
(R n ) ⊥ = ⃗ 0
Similarly,
{ } ⊥
⃗ 0 =(R n )
Proof. Here,⃗0 is the zero vector of R n .Since⃗x •⃗0 = 0forall⃗x ∈ R n , R n ⊆{⃗0} ⊥ .Since{⃗0} ⊥ ⊆ R n ,the
equality follows, i.e., {⃗0} ⊥ = R n .
Again, since ⃗x •⃗0 = 0forall⃗x ∈ R n , ⃗0 ∈ (R n ) ⊥ ,so{⃗0} ⊆(R n ) ⊥ . Suppose ⃗x ∈ R n , ⃗x ≠⃗0. Since
⃗x •⃗x = ||⃗x|| 2 and⃗x ≠⃗0, ⃗x •⃗x ≠ 0, so⃗x ∉ (R n ) ⊥ . Therefore (R n ) ⊥ ⊆{⃗0}, and thus (R n ) ⊥ = {⃗0}. ♠
In the next example, we will look at how to find W ⊥ .
Example 4.145: Orthogonal Complement
Let W be the plane through the origin given by the equation x − 2y + z = 0. Find a basis for the
orthogonal complement of W.
Solution.
From Example 4.140 we know that we can write W as
⎧⎡
⎨
W = span{⃗u 1 ,⃗u 2 } = span ⎣
⎩
1
0
−1
⎤
⎡
⎦, ⎣
0
1
2
⎤⎫
⎬
⎦
⎭