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M340L FINAL EXAM Your name: FALL, 2013 Dr. Schurle Your UTEID:
Show all your work on these pages. Be organized and neat. Your work should be your own; there should be no talking, reading notes, checking laptops, using cellphones,....
THE EXAM HAS 10 PROBLEMS, SOME WITH SEVERAL PARTS.
- (10 points) Hermione started with a matrix A and did the row operations below, in the order given, and obtained the matrix
B =
.
- Interchange Rows 1 and 4.
- Subtract 3 times Row 1 from Row 4.
- Divide Row 4 by 2.
Find the matrix A, and then give the solution set of Ax = 0 in parametric vector form. DO NOT USE A CALCULATOR FOR ANY PART OF THIS PROBLEM!! SHOW ALL YOUR WORK STEP BY STEP!!
YOUR SCORE: /
- (12 points) Are the columns of the following matrix linearly independent? Do they span R^3? Show all your work and justify your answers.
- (6 points) Is the set H of all polynomials of degree at most three which have integer coefficients a subspace of the vector space P 3? Justify your answer.
- (9 points) A 22 × 17 matrix A has rank 16.
(a) Is there a vector b in R^22 for which Ax = b is inconsistent? Justify your answer.
(b) What is the maximum number of linearly independent vectors in Col A?
(c) What is the minimum number of vectors needed to span Nul A?
- (6 points) Consider the following five polynomials in the vector space P 3.
1 + 2t + 3t^2 − t^3 , 2 + 4t + 6t^2 − 2 t^3 , t − t^2 + 2t^3 , 3 + 10t + 5t^2 + 5t^3 , t^2 + 3t^3
(a) Are the five polynomials linearly independent? Justify your answer.
(b) Do the five polynomials span all of P 3? Justify your answer.
- (16 points) A vector space V has a basis B = {b 1 , b 2 , b 3 } and another basis C where
c 1 = b 1 + 2b 2 + b 3
c 2 = 2b 1 + 5b 2 + b 3 c 3 = −b 1 + b 2 − b 3 .
(a) Find the change of coordinates matrix from C to B.
(b) Write x as a linear combination of b 1 , b 2 , b 3 if [x]C =
.
(c) Find the change of coordinates matrix from B to C.
(d) Suppose T is a linear transformation from V to V whose matrix relative to B is
. Find the matrix for T relative to C. Do all the arithmetic.
- (10 points) Let W be the subspace of R^5 spanned by the orthogonal vectors
, and
(a) Find the vector in W that is as close as possible to y =
(b) Find a vector in R^5 that is orthogonal to the subspace W. Is this vector a basis for the orthogonal complement W ⊥^ of W? Justify your answer.
- (8 points) We want to find a, b, c so that the parabola y = a + bx + cx^2 passes through the points (0, 3), (1, 9), (2, 20), and (3, 40).
(a) Write down the system of linear equations that we’d like to solve.
(b) Find the least squares solution of the system in (a).
