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TEST 2
Math 205 11/18/11 Name: (^) ︸ ︷︷ ︸ by writing my name i swear by the honor code
Read all of the following information before starting the exam:
- Show all work, clearly and in order if you want to get full credit (matrices should be reduced into RREF with calculator and you can just show the output). I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct).
- Circle or otherwise indicate your final answers.
- Please keep your written answers brief; be clear and to the point. I will take points off for rambling and for incorrect or irrelevant statements.
- This test has 7 problems and is worth 100 points, It is your responsibility to make sure that you have all of the pages!
- Good luck!
1. (15 points) Given A =
. Find a basis for the following.
a. (5 pts) Col(A)
b. (5 pts) Nul(A)
c. (5 pts) Row(A)
2. (9 points) Prove (verify in generality) that H = span{ v~ 1 , ~v 2 ,... , ~vp} where each ~vi is in V is a
subspace of V.
3. (12 points) A =
[
2 k 0 2
]
a. (3 pts) What is the det(A)? b. (3 pts) For what k is A invertible? c. (3 pts) What are the eigenvalues of A? d. (3 pts) For what values of k will A be diagonalizable?
6. (13 points) Let A be an invertible, diagonalizable matrix (A = P DP −^1 ) where
P =
[
a b c d
]
, ad − bc = 1 and D =
[
λ 1 0 0 λ 2
]
Find the general form for A−k^ (Remark: A−k^ is just (A−^1 )k).
7. (27 points) Short answer.
a. (6 pts) A matrix is called full rank when its rank is as large as possible. Let A be an m × n matrix of full rank. Justify whether or not A~x = ~b will be consistent if:
b. (4 pts) Give an example of a subset H of some vector space V of your choosing that is NOT a subspace. Briefly explain why.
c. (4 pts) A is a 6 × 6 matrix with 4 eigenvalues. Two of the eigenspaces are two-dimensional. Is A always diagonalizable? Why or why not?
d. (5 pts) Suppose A is an n × n matrix. If rank(A) = n, what is the Col(A), Row(A), and the Nul(A).
e. (4 pts) Let A and B be two matrices where AB exists. Explain why Nul(B) is a subset of Nul(AB) (ie. explain why every vector in Nul(B) is also in Nul(AB).
f. (4 pts) Consider the linear transformation which sends a 2×2 matrix to its transpose. Without making the matrix of this transformation find an eigenvalue and describe the associated eigenspace.
8. (1 point) Bonus: In problem 6, if λ 1 = λ 2 what is the general form of A−k?
