APPM 3310: Matrix Methods — Exam #1 — Sept 30, 2009
On the front of your bluebook print (1) your name, (2) your student ID number, and (3) a grading
table. Show all work in your bluebook. A correct answer with no supporting work may receive no
credit while an incorrect answer with some correct work may receive partial credit. One page of
notes is permitted, but no other books or electronic devices are allowed.
Please sign your bluebook under the Honor Code to indicate that you have neither
given nor received unauthorized assistance on this exam.
1. (30 points) True/false questions. In each case, you must also give a short explanation of your
answer.
(a) If Aand Bare square matrices and AB =Ithen BA =I.
(b) If A=LU gives the LU-decomposition of A, then det(A) = det(L).
(c) If Aand Bare nonsingular n×nmatrices, (A+B)−1=A−1+B−1.
(d) If Ais a nonsingular n×nmatrix, (AT)−1= (A−1)T.
(e) If Ais a singular n×nmatrix then Ax=bhas infinitely many solutions.
2. (40 points) Consider the system of equations with Ax=bwith A=
1 2 3 4
−1−276
−1−221
and b
an arbitrary vector.
(a) Use Gaussian elimination to transform the augmented matrix (A|b) into row echelon form.
(b) Find the LU decomposition of A.
(c) What is rank(A)?
(d) Define “range” and find a basis for the range of A.
(e) Define “kernel” and find a basis for the kernel of A.
(f) Set b= (0,2k , 3)T. For what values of kdoes the system Ax=bhave a unique solution?
no solution? infinitely many solutions?
3. (30 points)
(a) Give the definition of “basis” of a vector space.
(b) Explain why the set of all continuous functions such that f(1) = 0 forms a (vector)
subspace of the continuous functions, but the set of functions such that f(0) = 1 does
not.
(c) Let R⊂ Mm×nconsist of those matrices that have rank 2. Is Ra vector subspace of
Mm×n? Why or why not.
(d) Are the vectors 1, 2x−1, (x−1)2,x2∈P(2) independent? Why or why not?
(e) Suppose S= span(1,2x−1,(x−1)2, x2). What is the dimension of S?
(f) Find a basis for S.