MATH 4140/7140 Homework 3 Due 7/6/2016 Do all problems on different paper. I will not grade anything written on this page. Show your work. Answers without work will be given little or no credit. 1. You've seen that to determine if n vectors in the n-dimensional space Rn are linearly independent, it suffices to check if the matrix whose column vectors are these vectors is nonsingular. This technique, however, has to be adjusted when checking linear independence in other vector spaces. For instance, C (n1) [a, b] is the vector space consisting of all functions f (x) such that f (n1) (x) is continuous. In this case, C (n1) [a, b] is an infinite-dimensional space and while each f (x) is a vector, it is not a column vector in the sense we're accustomed to. Still, determinants can be used to check if a set of functions is linearly independent or not. Read about the Wronskian on pp. 138-140 in the book and see the examples there. Then determine if the following functions are linearly independent or not by using the fact that a collection of vectors is linearly independent in a subspace S of V if and only if they are linearly independent in V . (a) x + 1, x + 2, x2 1 in P3 . (b) cos x, sin x in C[0, 1]. (Hint: use a trig identity) (c) ex , ex , e2x in C(, ). (Hint: remember that you can alternatively use elementary column operations when finding the determinant) (The Wronskian is useful in checking the linear independence of solutions to differential equations.) 2. Let 2 2 3 1 , v2 = 2 , v3 = 2 . v1 = 2 0 2 Find the dimension of the subspace of R3 spanned by these vectors. 3. Let 3 1 3 4 A = 1 2 1 2 . 3 8 4 2 (a) Find a basis for the row space of A. (b) Find a basis for the column space of A. (c) Find a basis for the nullspace of A. (d) Compute the rank of A. 1 4. Let 1 1 1 1 , v 2 1 , v3 = 0 . v1 = 1 0 0 Observe that v1 , v2 , v3 are linearly independent, hence form a basis for R3 (you do not have to show this). Let F1 = {v1 , v2 , v3 }, F2 = {v2 , v1 , v3 }, F3 = {v3 , v2 , v1 } be ordered bases, and let S = {e1 , e2 , e3 } be the standard basis for R3 . Let x = (1, 3, 2)T . (a) Find the transition matrix MFS1 from the standard basis S to the basis F1 . (b) Find [x]F1 , i.e., find the coordinate vector of x with respect to the basis F1 . (c) Find the transition matrix MFF21 from the basis F1 to the basis F2 . (d) Find the transition matrix MFF31 from the basis F1 to the basis F3 . (e) Find the transition matrix MFF32 from the basis F2 to the basis F3 . 5. Let A and B be 7 4 matrices. Suppose that dim N (A) = 3 and the rank of B is 2. (a) What is the rank of A? (b) What is the dimension of the column space of A? (c) What is the dimension of N (B)? 2