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Spring 2012, Math 54
GSI: Shishir Agrawal
Discussions 106 and 107
Worksheet 5
Problem 1. Let P2 be the vector space of polynomials of degree at most 2. Let S
standard basis for P2 , and let B t1, 1 t, 1 2t 2t2 u.
t1, t, t2 u be the
(a) Show that B is a basis for P2 .
(b) Calculate the change-of-basis matrices PBÐS and PSÐB .
(c) Find the B-coordinate vector of the polynomial 2
2t2 .
4t
Problem 2. Let M2 be the set of all 2 2 matrices.
(a) Show that M2 is a vector space (under matrix addition and scalar multiplication).
(b) What is the dimension of M2 ? Give a basis for M2 .
1 2
and then let V be the the set of all 2 2 matrices A such that AB
2 4
is a subspace of M2 .
(c) Let B
0. Show that V
(d) What is the dimension of V ? Find a basis for V .
Solution. For part (b), observe that
"
1
0
is a basis for M2 , so dim M
2 4.
a b
For part (d), let A . Then AB
c d
if and only if a
only if
a
c
2b 0 and c
b
d
0
0
,
0
0
0
0
,
0
0
*
0
1
0 if and only if
1
2
1
0
,
0
1
2
4
a
c
2b
2d
2a
2c
4b
4d
0
0
0
0
2d 0, if and only if a 2b and c 2d. In other words, AB
A
2b
2d
b
d
b 02
1
0
c
0
2
0
1
for arbitrary real numbers b and c. Thus V is spanned by the set
"
2
0
1
,
0
0
2
*
0
1
which are clearly linearly independent. Thus this is a basis and dim V
Problem 3.
1 2
Let A 0 1
1
1
0
2.
0
(a) Find the characteristic polynomial of A.
1
2.
0 if and
(b) Find the eigenvalues of A.
(c) Is A diagonalizable?
Problem 4. Determine if each of the following statements are true or false.
(a) Every square matrix has at least 1 real eigenvalue.
(b) Every 3 3 matrix has at least 1 real eigenvalue.
(c) The sum of two eigenvalues of a matrix A is an eigenvalue of A.
(d) The sum of two eigenvectors of a matrix A is an eigenvector of A.
Solution.
False: polynomials of even degree may not have real roots.
True: cubic polynomials must have a real root by the intermediate value theorem.
False: almost anything is a counterexample.
False: take two eigenvectors corresponding to different eigenvalues to get a counterexample.
Problem 5. Let C 8 be the vector space of all infinitely differentiable functions f : R Ñ R, and let
T : C 8 Ñ C 8 be the linear transformation defined by T pf q f 1 . Using a bit of calculus, find the
eigenvalues and associated eigenspaces of T .
Solution. A real number λ is an eigenvalue of T if and only if there is a nonzero function f such that
f 1 λf , if and only if f 1 pxq{f pxq λ, if and only if log f pxq λx d for some constant d, if and only
if f pxq ceλx for some constant c. In other words, every real number λ is an eigenvalue of T , and the
eigenspace associated to λ is the one-dimensional subspace of C 8 spanned by the infinitely differentiable
function f pxq eλx .
2