EXAM
Exam 3
Takehome Exam
Math 2360–202, Summer II, 2014
August 1, 2014
• This is a Take-home exam.
• Write all of your answers on separate sheets of paper.
You can keep the exam questions. You must show
enough work to justify your answers. Unless
otherwise instructed, give
√ exact answers, not
approximations (e.g., 2, not 1.414).
• Unless otherwise instructed, you can use a calculator
to do the computations according the the alogrithms
taught in class. State clearly what you are going to
put into the calculator and what you got out.
• You can use the textbook and your notes. You can
discuss the problems with other people, but write up
your own answers, don’t just copy from someone else.
• This exam has 6 problems. There are 450 points
total. Good luck!
70 pts.
Problem 1.
Consider the space
P3 = {a2 x2 + a1 x + a0 | ai ∈ R},
of polynomials of degree less than three. One basis of P3 is
P = 1 x x2 .
Define polynomials by
q1 (x) = 1,
q2 (x) = x − 2,
q3 (x) = (x − 2)2 = x2 − 4x + 4,
and define
Q = q1 (x) q2 (x) q3 (x) .
A. Verify that Q is a basis and find the change of basis matrices SPQ and SQP .
B. Let f (x) = 2 + 3x + 5x2 . Find the coordinates [f (x)]P .
C. Find the coordinates [f (x)]Q of f (x) with respect to the basis Q. Write f (x)
as a linear combination of q1 (x), q2 (x) and q3 (x). Expand and simplify this
expression to check that it’s correct.
D. Let
g(x) = 3 + 2(x − 2) − 3(x − 2)2 .
Find the coordinates [g(x)]P of g(x) with respect to the basis P by using
the change of basis matrix. Check that this is correct.
1
80 pts.
Problem 2.
Consider the vector space C ∞ (R) of infinitely differentiable functions on R.
A. Show that the functions ex and e−x are linearly independent in C ∞ (R) by
using the Wronskian.
∞
x
−x
B. Let S be
xthe two
dimensional subspace of C (R) spanned by e and e ,
−x
e
so U = e
is a basis of S.
Recall that hyperbolic sine and cosine are defined by
ex + e−x
ex − e−x
, sinh(x) =
.
2
2
Let H = cosh(x) sinh(x) . Find the matrix A so that H = UA and show
that H is a basis of S. Find the transition matrices SU H and SU H .
cosh(x) =
C. Let f (x) = 2ex − 5e−x . Use the transition matrices to find constants c1 and
c2 so that f (x) = c1 cosh(x) + c2 sinh(x).
Let g(x) = 5 cosh(x)+2 sinh(x). Use the transition matrices to find constants
a1 and a2 so that g(x) = a1 ex + a2 e−x .
D. Let D : S → S be the derivative operator D(f ) = f 0 . Find the matrix of D
with respect to the basis U. Use the transition matrices to find the matrix
of D with respect to the basis H. Using this information, what are the
derivatives of hyperbolic sine and cosine?
2
90 pts.
Problem 3. Recall that the standard basis of R2 is E = e1
1
0
e1 =
, e2 =
.
0
1
e2 where
Let
U = u1
where
u1 =
2
,
2
u2 ,
u2 =
3
,
2
and let
V = v1
where
v1 =
4
,
3
v2 ,
v2 =
5
.
4
Then U and V are ordered bases of R2 (you don’t need to check that).
Let L : R2 → R2 be the linear transformation such that
L(u1 ) = 2u1 − 3u2
L(u2 ) = −3u1 + 2u2 .
A. Find the transition matrices SEU , SEV , SU E , SVE , SU V and SVU .
B. Find [L]U U , the matrix of L with respect to the basis U.
C. Find [L]EE , the matrix of L with respect to the standard basis.
D. Find [L]VV , the matrix of L with respect to the basis V.
E. Let w ∈ R2 be the vector whose coordinate vector with respect to V is
−2
[w]V =
.
4
Find [L(w)]V , the coordinate vector of L(v) with respect to the basis V.
50 pts.
Problem 4. Let
"
4
−1
6
−1
A=
#
.
By hand computation, find the characteristic polynomial and the eigenvalues
of A. (Do not find any eigenvectors.)
3
80 pts.
Problem 5. In each part, you are given a matrix A and its eigenvalues. Find
a basis for each of the eigenspaces of A and determine if A is
diagonalizable. If so, find a diagonal matrix D and an invertible matrix P so
that P −1 AP = D.
A. The matrix is

3
−2

A=
 2
−2
−2


−1 

0 −1
0
and the eigenvalues are −1 and 2.
B. The matrix is

−10 −9

A=

0
−1
−9
−9
14


2 
.
14
and the eigenvalues are −1 and 2.
80 pts.
Problem 6. In each part, you are given a matrix A and its eigenvalues. Find
a basis for each of the eigenspaces of A and determine if A is
diagonalizable. If so, find a diagonal matrix D and an invertible matrix P so
that P −1 AP = D.
A. The matrix is


373
266
−912
−1124

 182

A=
 52

124
485
−1508
156
−483
76
−266

−624 


−182 

−371
and the eigenvalues are 1 ± 2i.
B. The matrix is

−3143

22634
−23311
137

 −1208 8689

A=
 −772 5549

−4004 28751
−8940

52 


33 

172
and the eigenvalues are 3 ± 2i
4
−5706
−29553