MATH 311
Exam 2
Fall 2000
Section 502
Solutions
P. Yasskin
1. (20 points) Consider the vector space P 2 of polynomials of degree t 2.
Consider the function of two polynomials
1
˜p, q ™
; pŸx qŸx dx
0
a. (10 pts) Show ˜p, q ™ is an inner product.
1
1
˜q, p ™
; qŸx pŸx dx ; pŸx qŸx dx ˜p, q ™
˜p, p ™
; pŸx dx u 0 and 0 only when pŸx 0 for all x.
˜p, aq
br ™ ; pŸx Ÿaq br Ÿx dx a ; pŸx qŸx dx b ; pŸx rŸx dx a˜p, q ™ b˜p, r ™
0
1
0
2
0
1
1
1
0
0
0
b. (10 pts) Find cos 2 where 2 is the angle between the polynomials
p 5x 2 3 and q 3x.
˜p, q ™
1
1
1
0
0
0
; pŸx qŸx dx ; Ÿ5x 2 3 Ÿ3x dx ; Ÿ15x 3 9x dx 15x 4 9x 2
4
2
1
0
15 9 33
4
2
4
1
1
1
2
2
2
4
2
5
3
˜p, p ™ ; pŸx dx ; Ÿ5x 3 dx ; Ÿ25x 30x 9 dx 5x 10x 9x
˜q, q ™
0
1
0
1
0
1
0
0
0
; qŸx 2 dx ; Ÿ3x 2 dx 3x 2
cos 2 ˜p, q ™
|p||q|
1
0
24
3
33
11
4 24 3
8 2
2. (40 points) Consider the vector space P 2 of polynomials of degree t 2.
Consider the bases
e1 1
e2 x
f1 1 x
f2 x
e3 x2
f 3 "x x 2
Consider the function L : P 2 v P 2 given by
LŸp 2 pŸ0 pŸ1 x
a. (5 pts) Show L is linear.
LŸap bq 2Ÿap bq Ÿ0 Ÿap bq Ÿ1 x aŸ2 pŸ0 pŸ1 x bŸ2 qŸ0 qŸ1 x aLŸp bLŸq b. (5 pts) Find the matrix of L relative to the e-basis. Call it
L Ÿe 1 L Ÿ1 L Ÿe 2 L Ÿx L Ÿe 3 L Ÿx
2
2 x 2e 1 e 2
x
e2
x
e2
A.
ese
2 0 0
A
ese
1 1 1
0 0 0
1
c. (10 pts) Find the change of basis matrices
S
C from the f-basis to the e-basis, and
esf
f1 1 x
e1 e2
e2
"e 2 e 3
f2 x
f 3 "x x 2
S
1 0
C
esf
0
1 1 "1
0 0
1
C from the e-basis to the f-basis.
f se
1 0
0
1 0 0
1 1 "1
0 1 0
0 0
0 0 1
1
1 0 0
1 0 0
0 1 0
"1 1 1
0 0 1
0 0 1
R2 " R1 R3
1 0 0
C
f se
"1 1 1
0 0 1
d. (10 pts) Find the matrix of L relative to the f-basis. Call it B .
f sf
B f sf
C
f se
A
ese
C
esf
1 0 0
2 0 0
1 0
0
"1 1 1
1 1 1
1 1 "1
0 0 1
0 0 0
0 0
2 0 0
1
0 1 0
0 0 0
e. (5 pts) Find B by a second method.
f sf
Recall LŸp 2 pŸ0 pŸ1 x
L Ÿf 1 L Ÿ1 x L Ÿf 2 L Ÿx L Ÿf 3 L Ÿ" x x 2 2 2x 2f 1
x
f2
0
0
2 0 0
B f sf
0 1 0
0 0 0
f. (5 pts Extra Credit) What are the eigenvalues and corresponding eigenpolynomials of L?
(This required no computations.)
Since LŸf 1 2f 1 , we conclude f 1 is an eigenvector with eigenvalue 2.
Since LŸf 2 f 2 , we conclude f 2 is an eigenvector with eigenvalue 1.
Since LŸf 3 0, we conclude f 3 is an eigenvector with eigenvalue 0.
2
"6 "8
3. (40 points) Consider the matrix A 4
.
6
a. (15 pts) Find the eigenvalues and eigenvectors of A.
"6 " 5
"8
4 6"5
detŸA " 51 Ÿ"6 " 5 Ÿ6 " 5 32 5 2 " 4
"8 "8 0
5 2:
4
4 0
"4 "8 0
5 "2:
4
1 1 0
®
1 2 0
®
8 0
x
®
0 0 0
"t
y
x
®
0 0 0
5 2, "2
®
t
"2t
y
t
t
t
"1
1
"2
1
b. (10 pts) Find the diagonalizing matrix X so that A XDX "1 where D is diagonal.
What are D and X "1 ?
"1 "2
X
1
2
D
1
0
1
X "1 0 "2
2
"1 "1
c. (5 pts) Find A 10 .
"1 "2
A 10 XD 10 X "1 1
2 10
0
1
"1 "1
0 2 10
1
2
2 10
0
0 2 10
d. (5 pts) Find e At .
"1 "2
e At Xe Dt X "1 1
"1 "2
1
e 2t
1
0 e "2t
1
e 2t
2e 2t
"e "2t "e "2t
1
0
2
"1 "1
"e 2t 2e "2t "2e 2t 2e "2t
2e 2t " e "2t
e 2t " e "2t
e. (5 pts Extra Credit) Find sin = A .
4
3
5
7
HINT: sin x x " x x " x C
3!
5!
7!
sin = A
4
"1 "2
1
1
1
sin = 2
4
"1 "2
X sin = D X "1 4
1
0
0 "1
0 sin " = 2
4
1
1
0
2
"1 "1
"1 "2
1
1
1
2
"1 "1
1 2
1 1
"3 "4
2
3
3
4. (10 points) Let V Span£ v 1 , v 2 ¤ where
v 1 1
0
0
1
and v 2 1
0
0
1
1
0
.
Notice that V is a subspace of R 5 .
a. (6 pts) Find V µ the orthogonal subspace to V.
x R 5 3 v 1 x 0 and v 2 x 0
v 2 x b d 0
v 1 x a c e 0
Let x Ÿa, b, c, d, e T
Vµ "r " t
"s
a
b
1 0 1 0 1 0
Already reduced.
0 1 0 1 0 0
Vµ
x r
x c
r
d
s
e
t
"1
0
"1
"1
0
"1
0
"1
0
0
"1
0
s
1
0
t
0
Span
1
,
0
,
0
0
1
0
0
1
0
0
0
1
0
0
1
b. (2 pts) What is a basis for V µ ?
"1
0
"1
0
"1
0
1
,
0
,
0
0
1
0
0
0
1
c. (2 pts) What is the dimension of V µ ?
dim V µ 3
4