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M340L FINAL EXAM B
FALL, 2009
Dr. Schurle
9:00 SECTION, SATURDAY, DECEMBER 12
Your name:
Your UTEID:
Show all your work on these pages. If you use your calculator,
say so. Be organized and neat. Your work should be your own; there should be no
talking, reading notes, checking laptops, using cellphones,....
THE EXAM HAS 14 PROBLEMS.
- (6 points) Use parametric vector form to describe all solutions of the system Ax = 0 , where A is given below. DO NOT USE A CALCULATOR!! SHOW ALL YOUR WORK STEP BY STEP!!
A =
YOUR SCORE: /
- (7 points) Are the columns of the following matrix linearly independent? Do they span R^4? Show all your work and justify your answers.
- (7 points) Find the inverse of A =
[ 4 3 6 4
] and use it to solve 4AX + 3B = C for X,
where B =
[ 1 1 2 0 2 3
] and C =
[ 4 1 − 1 2 2 0
] .
- (6 points) The dimension of the column space of a 25 × 19 matrix A is 17.
(a) What is the largest number of linearly independent vectors in the row space of A?
(b) What is the smallest number of vectors needed to span Nul A?
(c) Is there a b in R^19 for which the system Ax = b is inconsistent? Justify your answer.
- (6 points) Suppose there are exactly 5 linearly independent solutions of a homogeneous system Ax = 0 of 32 equations in 37 unknowns. Does Ax = b have a solution for every b in R^32? Justify your answer.
- (5 points) Is −1 an eigenvalue of
?^ If so, find a basis for its
eigenspace. If not, explain why not. Show the work that justifies your answer.
- (15 points) This problem has five parts. (a) Show that B = { 1 , t, 1 + t + t^2 , 1 + t^3 } is a basis for the space P 3 of polynomials of degree at most three.
(b) If [r(t)]B =
, then find and simplify r(t).
(c) Find the change of coordinates matrix from B to the standard basis S = { 1 , t, t^2 , t^3 }.
(d) Find the change of coordinates matrix from the standard basis S to the basis B.
(e) Find [r(t)]B if r(t) = 2 + 3t + t^2 − t^3.
- (6 points) Let W be the subspace of R^5 spanned by the orthogonal vectors
and
, and let y =
(a) Find the vector in W that is closest to y.
(b) Find a nonzero vector in W ⊥.
- (6 points) Find an orthogonal basis for the column space of the matrix
.
