24 – 677 Name (Print): Fall 2019
Mid-term Exam 1 Practice
Time: 8:30 am to 10:20 am
This exam contains 8 pages (including this cover page) and 7 problems. Check to see if any pages are missing. Enter all requested information on the top of this page, and put your initials on the top of every page, in case the pages become separated.
You may use your equation sheet and calculator on this exam.
You are required to show your work on each problem on this exam. The following rules apply:
• Organize your work, in a reasonably neat and coherent way, in the space provided. Work scat- tered all over the page without a clear ordering will receive very little credit.
• Mysterious or unsupported answers will not receive full credit. A correct answer, unsup- ported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations will receive partial credit.
• If you need more space, use the back of the pages; clearly indicate when you have done this.
Do not write in the table to the right.
Problem
Points
Score
1
10
2
10
3
10
4
10
5
15
6
15
7
15
Total:
85
24 – 677 Mid-term Exam 1 Practice – Page 2 of 8
1. Please state whether the following statement is True or False. Explanation is not required.
(a) (3 points) The set of positive definite matrices forms a subspace [True or False].
(b) (3 points) The set of upper triangular matrices forms a subspace [True or False].
(c) (4 points) Let L : X → Y be a linear operator where (X , F ) is a 6-dimensional vector space and (Y , F ) is a 9-dimensional vector space. Then the nullity of L plus the rank of L equals 9.
[True or False].
24 – 677 Mid-term Exam 1 Practice – Page 3 of 8
2. Choose the best answer from the following multiple choices. Suppose that A = M1AˆM2, where
M1M2 = M2M1 = I4×4, and
3000 5−100 0−11 1
0 0 1 0 0 1 1 1 −1 −5 5 5 Aˆ = , M 1 = , M 2 =
0001 0 2 10 2 11−10−11
0 0 0 0 0 −1 0 1 −1 −5 5 6
(a) (5 points) Based on the above data, select the correct answer concerning the eigenvec- tors of A. (Just to be extra clear, the problem concerns right eigenvectors and not left eigenvectors.)
(A) All of the columns of M1 are eigenvector of A.
(B) Only two of the columns of M1 are eigenvector of A. (C) All of the columns of M2 are eigenvector of A.
(D) Only three of the columns of M2 are eigenvector of A. (E) None of the above.
(b) (5 points) Based on the above data, select the correct answer concerning the eigenvec- tors of A3. (Just to be extra clear, the problem concerns right eigenvectors and not left eigenvectors.)
(A) All of the columns of M1 are eigenvector of A3.
(B) Only two of the columns of M1 are eigenvector of A3. (C) All of the columns of M2 are eigenvector of A3.
(D) Only three of the columns of M2 are eigenvector of A3. (E) None of the above.
24 – 677 Mid-term Exam 1 Practice – Page 4 of 8 3. (10 points) Consider the linear algebraic equation:
2 −1 −1 −3 3x=0
−12 −1
Does a solution x exist? If so, is it unique? Does a solution exist if y = 1? 1
1
24 – 677 Mid-term Exam 1 Practice – Page 5 of 8 4. (10 points) For the matrix A given as
Calculate A2018.
1 1 0 A=0 0 1
001
24 – 677 Mid-term Exam 1 Practice – Page 6 of 8 5. (15 points) Find the SVD of the matrix A given as
1 0 1 0 A=0101
24 – 677 Mid-term Exam 1 Practice – Page 7 of 8
6. (15 points) Find the first order (linear) polynomial “closest” to the polynomial x2 using the
inner product ⟨f,g⟩ = 1 fgdx. In other words, find the projection of x2 onto the subspace of 0
1st order polynomials.
24 – 677 Mid-term Exam 1 Practice – Page 8 of 8
7. (15 points) Suppose that A is a 4 × 4 real matrix with characteristic polynomial det(λI − A) =
λ4 − 2λ2 + 4. Suppose further that b is a 4 × 1 real vector such that the set {A4b, A3b, A2b, Ab}
is linearly independent in (R4,R). What is the representation of A with respect to the basis {A4b, A3b, A2b, Ab}?
Pay attention to the order of the vectors. Also, b is not an element of the given basis; that is not a typo.