Worksheet 11: Orthogonal Projections and Orthonormal Bases

Orthogonal Projections and Orthonormal Bases
1. If ~u1 , . . . , ~um are orthonormal vectors in Rn , must they be linearly independent? How can you be sure?
2. Let V be a subspace of Rn . For any vector ~x ∈ Rn , we can write ~x = ~xk + ~x⊥ where ~xk is in V and
~x⊥ is orthogonal to V .(1) Then, ~xk is called the orthogonal projection of ~x onto V and denoted by
projV (~x).
Suppose we have an orthonormal basis (~u1 , . . . , ~um ) of V ; that is, (~u1 , . . . , ~um ) is a basis of V with the
property that ~u1 , . . . , ~um are orthonormal.
(a) Explain why projV (~x) can be written as projV (~x) = c1 ~u1 + · · · + cm ~um for some scalars c1 , . . . , cm .
(b) Since ~x = ~xk + ~x⊥ , we can use (a) to write
~x = (c1 ~u1 + · · · + cm ~um ) + ~x⊥ .
Express the coefficient ck in terms of ~x, ~u1 , . . . , ~um .
(c) Write a formula for projV (~x) in terms of ~x, ~u1 , . . . , ~um .
(d) In coming up with this formula for projV (~x), where was it important that (~u1 , . . . , ~um ) be an
orthonormal basis of V ?
(1) When
we say ~
x⊥ is orthogonal to V , we mean that ~
x⊥ is orthogonal to every vector in V .
1




 
1/3
−2/3
1
3. Let V be the plane 2x + 2y + z = 0, ~u1 = −2/3, and ~u2 =  1/3. Let ~x = 4.
2/3
2/3
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(a) Verify that (~u1 , ~u2 ) is an orthonormal basis of V .
(b) Find projV (~x). (Check that your answer is reasonable by computing the difference ~x − projV (~x).
What should be true about this vector?)
4. Let V be an m-dimensional subspace of Rn . Consider the linear transformation projV : Rn → Rn .
(a) What is im projV ? What is rank projV ?
(b) What is ker projV ? What is its dimension?
2
5. In each part, you are given a subspace V of some Rn . Describe V ⊥ .
(a) y = 3x in R2 .
(b) y = 3x in R3 .
 
1
(c) span 2.
3
6. Suppose B = (~u1 , ~u2 , ~u3 ) is an orthonormal basis of R3 . In each part of this problem, you are given the
B-matrix of a linear transformation T : R3 → R3 . Describe the linear transformation geometrically
(be as specific as you can).




0 −1 0
1 0 0
0 0
(a) 1
(b) 0 0 0
0
0 1
0 0 1
If the basis B were not orthonormal, how would your answers change?
3
7. Let B = (~u1 , . . . , ~un ) be an orthonormal basis of Rn , and suppose [~x]B
 
c1
 .. 
=  . .
cn
(a) Let V be the subspace of Rn spanned by ~u1 , . . . , ~um . What is [projV (~x)]B ?
(b) What is k~xk in terms of c1 , . . . , cn ?
4