Homework 11
Due: Wednesday, December 3, 2008
1. Which of the following define inner products on R3 ? Justify. (In each, x and y are arbitrary
elements of R3 , with x = ( x1 , x2 , x3 )T and y = ( y1 , y2 , y3 )T .)
(a) h x, yi = x1 y1 + x3 y3
(b) h x, yi = x1 y1 − x2 y2 + x3 y3
(c) h x, yi = 2x1 y1 + x2 y2 + 4x3 y3
(d) h x, yi = x1 y21 + x22 y22 + x23 y23 .
2. Consider R3 with its standard inner product. Determine which of the following sets of vectors are orthonormal. Show enough work to make clear that you checked the necessary
properties. (H INT: Two are orthonormal.)
     √ 
3
0


 1

√
3√




2
3


0 ,  √2  , − 3 
(a)
√


 0

2
3
2
3
    

0
0



 1
 √2   √2 


0 ,  √2  , −√ 2 
(b)



 0
2
2
2
2

  √   √

35
14



 √0
 10   3√735   37√14 
,
,
(c)  √
 −√ 14 
10   √
35



 3 10
− 35
14
10
35
14
 
3

3. For each part of #2 which is indeed orthonormal, write 4 as a linear combination of the
1
basis elements without solving a system of linear equations. Please show your work.
4. On P2 (R)[ x], consider the inner product given by
h p, qi =
Z 1
p( x)q( x)dx.
0
Apply the Gram-Schmidt procedure to the basis {1, x, x2 } to produce an orthonormal basis
of P2 (R)[ x]
5. Suppose that U is a subspace of an inner product space V, and let
U ⊥ = {v ∈ V : hu, vi = 0 for all u ∈ U }
be the orthogonal complement of U. Prove that U ⊥ is a subspace of V.
Professor Jeff Achter
Colorado State University
M369 Linear Algebra
Fall 2008