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Problem Set 5
Math 423, Section 200, Spring 2015
Due: Friday, March 6.
Review Sections 6A and 6B in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Friday, March 6. One of the problems is considered extra credit.
Remember to fully justify all your answers, and provide complete details. Neatness is greatly
appreciated.
1.
a. Show that the function R2 × R2 → R that takes ((x1 , x2 ), (y1 , y2 )) to |x1 y1 | + |x2 y2 | is not
an inner product on R2 .
b. Show that the function R3 × R3 → R that takes ((x1 , x2 , x3 ), (y1 , y2 , y3 )) to x1 y1 + x3 y3 is
not an inner product.
2. Let T ∈ L (V),√where V is an inner product space. Suppose that kT vk ≤ kvk for all v ∈ V.
Prove that T − 2 id is invertible.
3. Suppose u, v are elements of an inner product space V. Prove that hu, vi = 0 if and only if
kuk ≤ ku + λvk
for all λ ∈ F.
4. Let u, v ∈ V, where V is an inner product space. Suppose that kuk = kvk = hu, vi = 1. Prove
that u = v.
√
5. Let V be a complex inner product space. Denote i = −1. Prove that for all u, v ∈ V
hu, vi =
ku + vk2 − ku − vk2 + ku + ivk2 i − ku − ivk2 i
4
6. Suppose that T ∈ L (R3 ) has an upper triangular matrix with respect to the basis (1, 0, 0), (1, 1, 1), (1, 1, 2).
Find an orthonormal basis of R3 (with the Euclidean inner product) with respect to which T
has an upper triangular matrix.
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7. On P2 (R) consider the inner product given by
Z 1
hp, qi =
p(x)q(x)dx
0
(you do not need to prove that this is an inner product). Apply the Gram-Schmidt procedure
to 1, x, x2 to produce an orthonormal basis of P2 (R).
8. Find an orthonormal basis of P2 (R) (same inner product as in the previous exercise) such that
the differentiation operator on P2 (R) has an upper triangular basis with respect to this matrix.
9. Find a polynomial q ∈ P2 (R) such that
p
1
2
=
Z
1
p(x)q(x)dx
0
for all p ∈ P2 (R).
10. Suppose that h·, ·i1 and h·, ·i2 are two inner products on a finite dimensional vector space V
such that hu, vi1 = 0 if and only if hu, vi2 = 0. Prove that there exists a positive number c such
that hu, vi1 = chu, vi2 for all u, v ∈ V.
11. Suppose that v1 , . . . , vm ∈ V are linearly independent. (V is an inner product space.) Show that
there exists w ∈ V such that hw, v j i > 0 for all j = 1, . . . , m.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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