S OCIAL E COLOGY W EEK 9: E IGENVALUES AND EIGENVECTORS
M ATH 121A W INTER 2017
1. a. Prove that the vectors {(1, 0), (0, 1)} form a basis of C2 as a C-vector space.
b. Prove that the vectors {(2, 1
3i), (1 + 3i, 5)} form a basis of C2 as a C-vector space.
c. Consider the C-linear transformation T : C2 ! C2 given by (x, y) 7! (3x 2y, 5x + 5y).
What is the matrix of this linear transformation relative to the ordered basis {(1, 0), (0, 1)}?
d. Let T be as above. What is the matrix corresponding to T relative to the ordered basis
{(2, 1 3i), (1 + 3i, 5)}?
2. Assume T : V ! V has an eigenvector v corresponding to eigenvalue . Find an eigenvector
of T 3 and a corresponding eigenvalue.
3. Let A 2 M3⇥3 (R) denote the matrix
0
a 3
@ 0 b
0 0
1
5
4 A,
c
where a, b, c 2 R.
a. Find an eigenvector of A corresponding to the eigenvalue a.
b. Assume a 6= b. Find an eigenvector of A of the form (?, 1, 0) corresponding to the eigenvalue
b.
c. Assume none of a, b, c is equal to 1. Prove that ~0 2 R3 is the only vector fixed by A.
4. Find all eigenvalues of T : P (R) ! P (R), where T corresponds to differentiation.
5. Let V be a finite-dimensional vector space and let T : V ! V be a linear transformation. Prove
that T is invertible if and only if 0 is not an eigenvalue of T .
6. Why is there a unique linear transformation T : R2 ! R2 that sends (1, 0) 7! (1, 0) and
(1, 1) 7! (12, 12)? What are the eigenvalues of this linear transformation?
7. Give an example of a 2 ⇥ 2 matrix in M2⇥2 (R) which has no eigenvectors in R2 .
8. Find all solutions to the equation
✓
0
1
a. where y,
b. where y,
2 R;
2 C.
1
0
◆✓
4
y
◆
=
✓
4
y
◆
,