Ok forget about the previous questions...

Can you help me on A10Q #5-8 just the answers

A10W #2 and 3

 

Math 636 Assignment 10 - Quiz Component 1. Consider P2 (R) with inner product hp(x), q(x)i = p(0)q(0) + p(1)q(1) + p(2)q(2).
Find the value of h1 + x − 2x2 , x + x2 i.
(a) h1 + x − 2x2 , x + x2 i = 30
(b) h1 + x − 2x2 , x + x2 i = −28
(c) h1 + x − 2x2 , x + x2 i = −30
(d) h1 + x − 2x2 , x + x2 i = 31


1
3
√
2. Let A =
∈ M2×2 (R). Then
− 2 1
p
√
√
(a) kAk = 9
(b) kAk = 3
(c) kAk = 5 + 2
(d) kAk = 13


3 2
3. In M2×2 (R), which of the following matrices is not orthogonal to B =
.
−1 2








1 −3
3 −2
2 −2
0 0
(a)
(b)
(c)
(d)
2 2
1 −2
4 1
0 0
4. Which of the following statements is false.
(a) If P is orthogonal, then P is invertible.
(b) If det P = ±1, then P is an orthogonal matrix.
(c) If P is an orthogonal matrix, then det P = ±1.
(d) If P is orthogonal, then Col(P ) = Row(P ).
For questions 5 - 8, determine if the statement is True or False. You should make sure that
you have a proof of each true statement and a counter example for each false statement.
5. The inner product of two vectors cannot be negative.
(a) True.
(b) False.
6. Let W be a subspace of an inner product space V. If ~u, ~v ∈ W⊥ , then ~u + ~v ∈ W⊥ .
(a) True.
(b) False.
7. If {~v1 , . . . , ~vn } is a basis for an inner product space V and ~x ∈ V such that h~x, ~vi i = 0
for 1 ≤ i ≤ n, then ~x = ~0.
(a) True.
(b) False.
8. If {~v1 , . . . , ~vn−1 } is orthonormal in an n-dimensional inner product space V, then there
is a unique vector ~vn ∈ V such that {~v1 , . . . , ~vn−1 , ~vn } is orthonormal.
(a) True.
(b) False. 1

Math 636 - Assignment 10 - Written Component
Due: Friday, July 22 at 4:00PM 1. Consider the function on R2 defined by
   
v1
w
, 1
= 2v1 w1 − v1 w2 − v2 w1 + v2 w2
v2
w2
(a) Prove that this function is an inner product on R2 .
(b) Show that the standard basis vectors ~e1 , ~e2 for R2 are not orthogonal under this inner product.
(c) Find a basis for R2 that is orthogonal under this inner product.

 
 
 

1 1
−1 1
4 −2
1 3
2. Let B =
,
,
,
be a spanning set for a subspace S of M2×2 (R).
1 1
1 1
−2 −2
3 1
Use the Gram-Schmidt procedure on B (in this order) to find an orthonormal basis for S.
3. Let A and B be n × n matrices. If there exists an orthogonal matrix P such that P T AP = B, then
we say that A and B are orthogonally similar.
(a) Prove that if A and B are orthogonally similar and are both invertible, then A−1 and B −1 are
also orthogonally similar.
(b) Prove that if C and D are orthogonally similar, then C 2 and D2 are also orthogonally similar.
4. Let V be an n-dimensional real inner product space and let h~x, ~y i and [~x, ~y ] both be two different
inner products on V Prove that there exists a linear mapping L : V → V such that
[L(~x), L(~y )] = h~x, ~y i, 1 for all ~x, ~y ∈ V

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