MATH 212 Wells College Linear Algebra Questions

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