The work has to do with basis, eigenvalues, eigenvector, transformation, diagonalizable, etc

MATH 212

FINAL

Spring 2020

Name

• No calculators or computers!

• Attach solutions!

• Show your work!

Due: Friday, May 8th, by 11:59 PM (or before!)

−2 0 −9

1. (5 points) Let B = 0 3 0 . Find all the eigenvalues of B.

0 5 6

2. (20 points)

(a) (12 points) Decide whether each of the following matrices is diagonalizable.

Justify your answer in each case. You do not need to find P , D, or P −1 .

1 7 0

A = 0 3 2 ,

0 0 0

(b) (8 points) Find E

2020

, where E =

1 0

2 −1

1 2 1

B = 0 1 0 .

0 2 2

.

3. (10 points) Consider the 2 × 2 matrix

B=

a b

c d

where a, b, c, d are real numbers, and a 6= 0. Find all values d such that rank(B) = 1.

4. (20 points)

(a) (10 points) Is the set of all vectors v ∈ Rn with ||v|| = 1 a subspace of Rn ?

Write your answer in complete sentences.

(b) (10 points) Find a vector q(t) such that the set

{2t + 1, t2 − 1, q(t)}

is a basis B for P2 . Justify your answer.

5. (22 points) Let S : P2 → P2 be defined as S(p(t)) := tp0 (t) + p0 (0) (where primes are

derivatives with respect to t).

(a) (10 points) Show that S is a linear transformation.

(b) (8 points) Find a basis B for ker(S).

(c) (4 points) Is S one-to-one?

Give a one sentence explanation of your answer.

−6 −3

6. (18 points) Let A = 11 5.5 , and let T be a linear transformation given by T (x) = Ax.

2

1

(a) (6 points) Describe the vectors in Nul(A) in parametric vector form.

(b) (6 points) Is T one-to-one? Explain.

(c) (6 points) Find three distinct non-zero vectors in Col(A).

7. (30 points) Let A =

0 1

.

1 0

(a) (5 points) Write the characteristic equation of A and find all eigenvalues of A.

(b) (10 points) Find a basis B for each eigenspace.

(c) (10 points) Is A diagonalizable? If yes, find P , P −1 , and D such that A = P DP −1 .

(d) (5 points) Find A3 using part (c).

8. (10 points) Let A =

5 1

. This matrix A has only one eigenvalue, λ = 5.

0 5

(a) (5 points) Find a basis B for the eigenspace corresponding to λ.

(b) (5 points) Is the matrix A diagonalizable? Explain.

9. (BONUS: [Up to] 20 points) For each subset H of the (appropriate) vector space V below,

if H is a subspace of V, find a basis B for it, or explain why there is no basis (e.g., maybe

the space has infinite dimension); if H is not a subspace of V, explain why not.

(a) (5 points) Let H be all polynomials in V = P4 which have no quadratic (x2 ) term.

3a + 2b

(b) (5 points) Let H be all vectors in V = R3 of the form 5ab , where a and b are

0

arbitrary.

3 5

(c) (5 points) Let H = Col(A), where A =

.

2 3

3 5

(d) (5 points) Let H = Nul(A), where A =

.

2 3

Please sign:

This exam is entirely my own work. I did not consult with anyone else for help

with the exam.