APPM 3310: Matrix Methods — Exam #1 — February 20, 2008
On the front of your bluebook print (1) your name, (2) your student ID number, and (3) a grading
table. Explain all of your answers. A correct answer with no supporting work may receive no
credit while an incorrect answer with some correct work may receive partial credit. No electronic
devices of any kind (e.g. cell phones, calculators, etc.) are permitted.
Please sign your bluebook under the Honor Code to indicate that you have neither given
nor received unauthorized assistance on this exam.
1. (40 points) For this problem, let A=
1 3 −5 2
2−1 1 −4
4 5 −9 0
(a) Find conditions on bso that Ax=bhas (i) one, (ii) none, or (iii) infinitely many solutions.
(b) Write your answer to part (a) in vector form, (i.e. b= (b1,b2, b3)T) in order to find a basis
for all of the bthat have a solution to Ax=b.
(c) Find a basis for ker(A) and rng(A).
(d) Is the subspace formed by the span of the vectors in part (b) the same as rng(A)? Show
this explicitly one way or the other.
(e) Find the LU decomposition of A.
2. (40 points) This question is worth 40 points. You must do part (a). Then, answer 3 of the
remaining 4 parts. (You can do all four for extra credit.)
(a) Carefully define each of the four fundamental subspaces of an m×nmatrix A. State the
Fundamental Theorem of Linear Algebra, part 1.
(b) Suppose Ais an m×nmatrix and Bis a p×qmatrix. If ker(A) = ker(B) show that
rank(A) = rank(B). (Hint: How does part (a) help you?)
(c) True or False (prove or find a counterexample): ker(A)⊂ker(A2) for a square n×nmatrix
A.
(d) (i) Describe three different ways you could tell whether a matrix is singular or nonsingular.
(ii) Now, suppose you know Ais n×nand Au=Avfor some u6=v. Is Asingular or
nonsingular? Explain.
(e) Give the definition of symmetric matrix. If Aand Bare both symmetric, which of the
following must be symmetric? (i)A2−B2; (ii) (A+B)(A−B); (iii)AB A.
3. (20 points) Let V= span{1, x2, x4}.
(a) Is Va subspace of P(4), where P(4) is the vector space of all polynomials of degree less than
or equal to 4?
(b) Are the polynomials p1(x) = 1 + x2,p2(x) = 1 −x2, and p3(x) = 1 −x4a basis for V?