Name
Math 311
Exam 2 Version A
Section 502
Points indicated. Show all work.
1.
Spring 2015
P. Yasskin
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15 points
Let P 5 be the vector space of polynomials of degree less than 5.
Consider the subspace V Span v 1 , v 2 , v 3 , v 4 where
v 1 2 3x 2 , v 2 x 3x 3 , v 3 x 2 x 3 x 4 , v 4 2 x 3x 4
Find a basis for V What is dim V ?
1
2.
38 points
Consider the vector space V Span 1, sin 2x , cos 2x
with the usual addition and
scalar multiplication of functions. Two bases are:
e 1 1 e 2 sin 2x e 3 cos 2x
and
E 1 sin 2 x E 2 cos 2 x E 3 sin x cos x
Note: You do NOT need to prove they are bases.
Hints: Here are some useful identities:
1 cos 2x
1 cos 2x
2 sin x cos x , cos 2x
cos 2 x sin 2 x , sin 2 x
, cos 2 x
sin 2x
2
2
a. (5) Find the change of basis matrix
C
from the e basis to the E basis by using the
E e
identities.
b. (5) Find the change of basis matrix
C
from the E basis to the e basis by using the
e E
identities.
c. (2) Verify
C
E e
and
are inverses.
C
e E
d. (4) For the function f
sin 2x
4 sin 2 x , what are its components
f
e
and
f E?
2
e. (5) Find the matrix
A
of the derivative operator D
d
dx
relative to the e basis.
of the derivative operator D
d
dx
relative to the E basis.
e e
f. (5) Find the matrix
B
E E
Do NOT use the change of basis matrices.
g. (2) A and B are related by a similarity transformation: B
SAS 1 . What is S?
h. (3) What is Im D ? Give a basis. What is dim Im D ?
HINT: Let f
a 1
b sin 2x
c cos 2x .
i. (3) What is Ker D ? Give a basis. What is dim Ker D ?
j. (2) Is D onto? Why or why not?
k. (2) Is D 1-1? Why or why not?
3
3.
25 points
Consider the vector space S of symmetric 2 2 matrices. The following are
symmetric, biinear forms on S. Which one(s) are inner products? Why or why not? You do not need
to check they are symmetic or bilinear, just that they are positive definite.
HINTS: Let A
a b
b d
. Compute
a. (9)
A, B
tr AGB
where G
b. (8)
A, B
tr AGB
where G
c. (8)
A, B
tr AGB
where G
A, A . Look for perfect squares or complete the squares.
1 1
1 1
3 1
1 1
1 2
2 1
4
4.
27 points
Consider the vector space S of symmetric 2
A, B
where G
tr AGB
4 0
0 1
V1
2 0
0 3
.
3 4
a. (8) Find the angle between the matrices A
b. (19) A basis for S is
2 matrices with the inner product
and B
4 3
V2
2 1
1 3
Apply the Gram-Schmidt Procedure to the V 1 , V 2 , V 3
W 1 , W 2 , W 3 and an orthonormal basis U 1 , U 2 , U 3 .
V3
4 3
.
3 4
1 0
0
1
.
basis to produce an orthogonal basis
5
6
7
5.
2
10 points EC
Consider the vector space V
x1, x2
x 1 0 and x 2 0
consisting of ordered pairs of positive numbers with addition and multiplication defined by
x1, x2
y1, y2
x 1 y 1 , x 2 y 2 and a
x1, x2
a
a
x1 , x2
So vector addition is real number multiplication of corresponding components and scalar
multiplication is real number exponentiation of each component. Note the zero vector is 0
a. (5) Is u 1
1, 2
and u 2
3, 1
a basis? Why or why not?
b. (5) Is v 1
1, 1
and v 2
3, 1
a basis? Why or why not?
1, 1 .
8