LINEAR ALGEBRA (Prof. De Loera) Graduate Proficiency Exam Fall 2005

Show all details when solving every problem. Be organized and use the notation appropiately. Write legibly. Solve the problems without any assistance, in particular no calculators.

1. Given the matrix A = [ \begin{matrix}2 & −3 & −7 & 5 & 2 \\ 1 & −2 & −4 & 3 & 1 \\ 2 & 0 & −4 & 2 & 1 \\ 1 & −5 & −7 & 6 & 2 \end{matrix} ] find a basis for (a) range(A), (b) kernel(A).

Determine whether the vector [2, 2, −3, 7] is in the range of A or not.

2. Let A = [ \begin{matrix}5 & −6 & −6 \\ −1 & 4 & 2 \\ 3 & −6 & −4 \end{matrix} ]. Find the eigenvalues of A and if possible find a matrix S such that

S^{−1}AS is a diagonal matrix.

3. Can the matrix

[ \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ] be unitarily diagonalized? If yes, do it!

4. Given the matrix [ \begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 6 & −11 & 6 \end{matrix} ] Find its eigenvalues, its characteristic polynomial and if pos-

sible diagonalize it (find explicit S such that S^{−1}AS is diagonal).

5. The linear transformation given by derivation d/dt acts on the vector space of all polynomials with degree at most 3. Represent this linear transformation on the basis given by {1, t, t^{2}, t^{3}}. What is the Jordan canonical form of this matrix?

6. (a) What are the possible Jordan forms of an n × n matrix A such that A^{3} = I?

(b) What are the possible Jordan forms of a matrix A ∈ M_{6} with characteristic polynomial p_{A}(t) = (t + 3)^{4}(t − 2)^{2}?

7. Apply the Gram-Schmidt procedure to the vectors x_{1} = (1, 1, 0), x_{2} = (2, 0, 1) and x_{3} = (2, 2, 1).

8. Prove or disprove: Let A = {x_{1}, x_{2}, …, x_{n}} be linearly independent vectors, then A ∪ {y} is a set of linearly independent vectors if and only if y ∉ span(A).

9. A linear transformation P is a projection if P^{2} = P. Prove

a) P is a projection if and only if I − P is a projection.

b) y ∈ range(P) if and only if P y = y.

c) range(P) = kernel(I − P).

d) the vector space V is the direction sum of range(P) and kernel(P).

10. (a) If Λ denotes the diagonal n × n matrix with eigenvalues λ_{1}, …, λ_{n}. Show that ΛB = BΛ for some B, n × n matrix, if and only if B is also diagonal.

(b) Suppose A ∈ M_{n} has n distinct eigenvalues. If AB = BA for some B ∈ M_{n}, show that A, B are simultaneously diagonalizable.

11. Let A ∈ M_{n}. If rank(A) = 1, then there exists a scalar β ∈ ℂ such that A^{2} = βA.

12. Given a 4 × 4 matrix of rank 3, explain which are the possible A-invariate subspaces of ℂ^{4}.

13. Are there real 3×3 matrix whose minimal polynomial is x^{2} +1? How about if the matrix is complex ? What happens if the matrix is 2 × 2?

14. True or false (justify):

a) Similar matrices have the same minimal and characteristic polynomial

b) If two matrices have the same minimal and characteristic polynomial they must be similar.

c) inverse of an upper triangular matrix is upper triangular.

d) Let W_{1}, W_{2} be subspaces of V. Then W_{1} ∩ W_{2} is a subspace too.

e) Let W_{1}, W_{2} be subspaces of V. Then W_{1} ∪ W_{2} is a subspace too.

f) Let A be a k × n matrix. dim(range(A)) + dim(Kernel(A)) = n

g) Let A ∈ M_{k,n} and B ∈ M_{n,k}. If tr(X) denotes the trace of X, then tr(AB) = tr(BA).

15. Let A, B ∈ M_{n}. Prove or disprove that a) range(A + B) ⊂ range(A) + range(B).

b) range(AB) ⊂ range(A).

c) rank(AB) ≤ rank(B).

d) If A is non-singular then rank(AB) = rank(B) = rank(BA).

16. Suppose A, B ∈ M_{n} commute and have eigenvalues a_{1}, …, a_{n} and b_{1}, …, b_{n} respectively. Explain, with a proof, what are the eigenvalues of AB

17. a) State definitions of normal matrix, nilpotent matrix, and Hermitian matrix.

b) Assume that A is normal and that A, B ∈ M_{n} commute. Prove that A^{∗} and B commute too. c)When is a normal matrix unitary? When is it Hermitian? When is it nilpotent?

d) State three conditions equivalent to normality.

18. Let W, Y be subspaces of V such that V = W + Y. If W ∩ Y = {0}, every vector v ∈ V has a unique representation in the form v = x + y with x ∈ W, y ∈ Y.

19. Let A ∈ M_{n} and x, y represent non-zero vectors such that Ax = 3x and Ay = 5y. Prove that x, y are linearly independent. How general is this fact?

20. Prove or disprove:

a) rank(A^{2}) ≤ rank(A).

b) A, B ∈ M_{n} with AB diagonalizable implies BA diagonalizable. c) If P is a projection its eigenvalues are only 0, 1.

d) Let P, Q be projections which are simultaneously diagonalizable, then P Q is a projection too.

21. Sally is playing “matrix of fortune” and knows that 4, −7, 2 are eigenvalues of the following matrix

[ \begin{matrix}2 & ∗ & ∗ & 6 \\ ∗ & 3 & ∗ & ∗ \\ ∗ & ∗ & −1 & ∗ \\ ∗ & ∗ & ∗ & 5 \end{matrix} ] To win the game she needs to find det(A). Please help her!

22. Prove or disprove (with justification):

a) If A, B are two n × n matrices and A = A^{t} and B = B^{t} then AB = (AB)^{t}.

b) If A is nilpotent matrix then 0 is the only eigenvalue of A.

c) Similar matrices have the same eigenvalues.

d) If λ is an eigenvalue of both A, B then λ is an eigenvalue of A + B.

e) Matrices with the same determinant have the same rank.

f) If λ is an eigenvalue of an orthogonal transformation then λ = ±1

g) Matrices with the same characteristic polynomial are similar.

h) The set of all n × n real matrices with determinant zero is a subspace of M_{n}.

i) If a square matrix A is diagonalizable so is A^{2}.

j) Similar matrices have the same eigenvectors.

k) If V, W are two distinct 2-dimensional subspaces of R^{3}, then V + W = R^{3}.

l) Let {x, y, z} be a linearly independent set of vectors in a vector space over a finite field. The set {x + y, x + z, y + z} is always linearly independent.

23. Let V and W be the following subspaces of ℝ^{4}.

V = span{(1, 3, −4, 7), (−1, 2, −1, 3)}, W = span{(2, 4, 3, 1), (2, 2, −1, 3)}. Determine bases for the subspaces V + W, W ∩ V, and V^{⊥}.

24. Given the matrix A = [ \begin{matrix}21 & 6 & −12 \\ 6 & 12 & −6 \\ −12 & −6 & 21 \end{matrix} ]

Find the eigenvalues and an orthonormal basis for each of the eigenspaces. Is this matrix diagonalizable over R? Can you find a matrix B, such that B^{2} = A.?

25. Determine whether the following matrices are diagonalizable over the indicated field. If possible find the diagonalization.

[ \begin{matrix}1 & −1 & 2 \\ 1 & −1 & 2 \\ 1 & −1 & 2 \end{matrix} ] over the real numbers

[ \begin{matrix}3 & 0 & 1 \\ −1 & 2 & −1 \\ 1 & 2 & 5 \end{matrix} ] over the real numbers

[ \begin{matrix}3 & −4 \\ 4 & 3 \end{matrix} ] over the complex numbers.

26. Let A be a normal and idempotent matrix. Then ∑_{i,j} |a_{ij}|^{2} = rank(A).

27. Let A = [ \begin{matrix}0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{matrix} ] . Find the Jordan canonical form of A.

28. Let A = [ \begin{matrix}3 & −3 & 6 \\ −3 & 2 & −9 \\ 6 & −9 & 2 \end{matrix} ] .

Is this matrix diagonalizable? If yes, please find the diagonalization. Show that A determines an indefinite quadratic form. Determine vectors x, y such that x^{t}Ax > 0 and y^{t}Ay < 0. How many eigenvalues are negative, positive, or zero?

29. For an m × n matrix A:

Prove that the following conditions are equivalent a) rank(A) = n, b) A^{t}A is invertible, c) BA = I for some n × m matrix B.

Prove that the eigenvalues of A^{t}A are non-negative and the same as those of AA^{t}.

30. Are the eigenspaces of a matrix always orthogonal to each other?