Rules
1. This is a take-home exam; the time of exam is Dec 1, 8:00AM to Dec 4, 8:00AM.
2. Solutions must be submitted by Dec 4, 8:00AM. Late submissions will not be accepted; there is no
exception.
3. Submissions must be in email attachment form, sent to . Only PDF/Word les
and scanned images are acceptable; scanned images must be clearly readable. Handwriting that is
hard to recognize may result in lowered score.
4. You may consult any publicly available written or online resources; however you must cite any result
you use, if it does not appear in the textbook, lecture notes or homework problems. Make sure that
the resource you cite contains the proof of the result you are using.
5. Collaboration is not allowed. You are also not allowed to discuss with anyone the exam problems, or
post them to Q&A websites, or online forums.
6. Justify your answer by writing out appropriate steps.
Problems
1. (12 points) Finish the following problems.
(a) Let (x1;y1) and (x2;y2) be two distinct points on the Euclidean plane. Prove that the equation
represents the line connecting (x1;y1) and (x2;y2).
(b) Let a linear mapping S : R4 ! R4. If rank(S) = 2 and ker(S) = ker(S2), prove that C4 =
ker(S) ran(S).
2. (24 points) Finish the following problems.
(a) Describe the possible Jordan normal forms for an 10 10 complex matrix A satisfying A4 = A2.
1
(b) Prove that there do not exist two n n matrices A and B such that AB BA = I.
(c) Let X be the complex linear space formed by complex polynomials of degree not exceeding n,
and D : p(x)7!p0(x) be a linear mapping from X to itself. Prove that D is nilpotent, and nd
the Jordan normal form. of D.
3. (24 points) Consider the matrix
(a) Find the eigenvalues of A, and the dimensions of all generalized eigenspaces.
(b) Write down the (lower triangular) Jordan normal form. J of A, and nd an invertible matrix P
such that A = PJP 1.
(c) Compute A1000.
4. (20 points) Given complex numbers a1; ;an, consider the circulant matrix
A =
0
BB
BB
B@
a1 a2 a3 an
an a1 a2 an 1
an 1 an a1 an 2
... ... ... ... ...
a2 a3 a4 a1
1
CC
CC
CA:
(a) Let z 2C, and v(z) = (1;z;z2; ;zn 1)T. Find the su cient and necessary condition for z,
such that v(z) is an eigenvector of A regardless of the choices of (a1; ;an); for such z, compute
the eigenvalue corresponding to v(z).
(b) Using results from part (a), compute the determinant of A.
5. (a) (20 points) Let A be a real n n matrix such that A2 + I = 0. Prove that n must be even, and
that A is similar, as a real matrix, to a quasi-diagonal matrix
(Hint: view A as a complex matrix and look for its complex eigenvalues and eigenvectors.)
(b) (10 points extra credit) State (without proof) the corresponding result to part (a) in the case
(A2 + I)m = 0; m 2:
6. (10 points extra credit) Let A be an n n invertible complex matrix. For each positive integer k, let
Ak be the kth power of A, and A[k] be a matrix such that each entry of A[k] is the kth power of the
corresponding entry of A. Suppose Ak = A[k] for each positive integer n, prove that A is diagonal.