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Exam 2 Math 314 Fall 2006 Name: Score: Instructions: Show your work in the spaces provided below for full credit. Use the reverse side for additional space, but clearly so indicate. You must clearly identify answers and show supporting work to receive any credit. Exact answers (e.g., π) are preferred to inexact (e.g., 3. 14 ). Point values of problems are given in parentheses. Notes or text in any form not allowed. Calculator is allowed.
(30) 1. Let A = [v 1 , v 2 , v 3 , v 4 ] =
(^) with reduced row echelon form R =
0 1 −^1414
(a) Find a basis for R(A), the row space of A.
(b) Find a basis for C(A), the column space of A.
(c) Find a basis for N (A), the null space of A.
(d) Find all possible linear combinations of the vectors v 1 , v 2 , v 3 , v 4 that sum to 0.
(e) Which vj 's are redundant in the list of vectors v 1 , v 2 , v 3 , v 4?
(16) 2. Use the Subspace Test to decide if W is a subspace of the vector space V , where (a) V = R^3 and W = {(a, b, a − b + 1) | a, b ∈ R}
(b) V = C[0, 1], the continuous functions on [0, 1] and W = {f (x) | f (x) ∈ C [0, 1] and f (1) = 0}.
(10) 3. Assume that 1 + x, x + x^2 , 1 − x is a basis of P 2 , the space of polynomials of degree at most two, and nd the coordinates of 2 + x^2 relative to this basis.
(16) 6. Fill in the blanks or answer True/False (T/F).
(a) Every vector space is nite dimensional (T/F).
(b) Elementary row operations on a matrix do not change the column space (T/F).
(c) If x = x 0 and x = x 1 are both vector solutions to the linear system Ax = b, then x 1 − x 0 is in the null space of A. (T/F).
(d) The function T : R^2 → R^2 given by T ((x, y)) = (x + y, x − 2 y) is linear (T/F) and one-to-one (T/F).
(e) The Basis Theorem asserts that every nite dimensional vector space.
(f) The Dimension Theorem asserts that.
(g) If A is a 5 × 5 matrix and det (A) = 2, then the rst 4 columns of A span a 4 dimensional subspace of R^5 (T/F).
(10) 7. (a) Show from de nition of linear dependence that the columns of the matrix
form a linearly dependent set.
(b) (Honors students only) Prove that any set of vectors v 1 , v 2 ,... , vn in a vector space V that contains the zero vector is a linearly dependent set.
