Math 18, Lecture A
Fall 2022
Section 6.2 - Orthogonal Sets
Definition. A subset {~u1 , ~u2 , . . . , ~up } of Rn is orthogonal if ~ui · ~uj = 0 whenever i 6= j.
     
1
1 
 0





1 , 0 , 0  is orthogonal.
Example. Show that


0
1
−1
Theorem. If S = {~u1 , ~u2 , . . . , ~up } is an orthogonal set of nonzero vectors in Rn , then S is
linearly independent.
Definition. Let W be a subspace of Rn . An orthogonal basis for W is a basis for W that is
also orthogonal.
      
1
1 
 0
Example. Explain why S = 1 , 0 ,  0  is an orthogonal basis for R3 .


0
1
−1
Theorem. Let {~u1 , ~u2 , . . . , ~up } be an orthogonal basis for a subspace W of Rn . Then for
~y · ~uj
~y · ~uj
each ~y ∈ W , if ~y = c1~u1 + c2~u2 + · · · + cp~up , then cj =
or
for j = 1, 2, . . . , p.
~uj · ~uj
k~uj k2
1
Math 18, Lecture A
Fall 2022
Section 6.2 - Orthogonal Sets
     
 
1
1 
1
 0
Example. β = 1 , 0 ,  0  is an orthogonal basis for R3 . Find [~y ]β if ~y = 2.


0
1
−1
3
Definition. A subset {~u1 , ~u2 , . . . , ~up } of Rn is orthonormal if it is orthogonal and every vector
~ui in the subset is a unit vector.
Let W be a subspace of Rn . An orthonormal basis for W is a basis for W that is also
orthonormal.
Example. Determine whether the basis for R3 is orthonormal.
1. {~e1 , ~e2 , ~e3 }
     
1
1 
 0





1 , 0 , 0
2.


0
1
−1
Definition. An orthogonal matrix is a square matrix U such that U T = U −1 .
2
Math 18, Lecture A
Fall 2022
Section 6.2 - Orthogonal Sets
Theorem. Let U be any matrix. Then
• U has orthonormal columns iff U T U = I.
• If U is square, then U is orthogonal iff its columns are orthonormal.
• If U is square, then U is orthogonal iff its rows are orthonormal.

0
Example. Determine whether 1
0
√1
2
0
√1
2
√1
2

0  is orthogonal.
− √12
Orthogonal Projections
Fix a nonzero vector ~u ∈ Rn . Can we decompose a vector ~y ∈ Rn into a sum of a scalar multiple
of ~u and a vector orthogonal to ~u?
~y = ŷ + ~z
3
Math 18, Lecture A
Fall 2022
Section 6.2 - Orthogonal Sets
Definition. Let ~u, ~y ∈ Rn where ~u 6= ~0. Let L be the line through ~0 and ~u. Then
• The orthogonal projection of ~y onto L is ŷ = projL ~y =
~y · ~u
~u.
~u · ~u
• The component of ~y orthogonal to L is ~z = ~y − projL ~y = ~y −
~y · ~u
~u.
~u · ~u
3
9
Example. Let ~u =
and ~y =
.
−9
5
1. Express ~y as a sum of two orthogonal vectors, one in Span{~u} and the other orthogonal to
~u.
2. Find the distance between ~y and the line L = Span{~u}.
4