Final Review Set - Math 240 (Spring 2017) Show all your work for each problem (you may use a calculator or computer - that is, \"technology\" as we would say - but explain what commands you used and your interpretation of the results. If possible, show screen shots of the application used.) (1) [8] Find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace? 1 ! = 1 , 1 3 2 ! = 3 , 1 4 0 ! = 1 , 3 2 1 3 ! = 4 , ! = 7 7 6 7 9 (2) [8] Use the property of determinants to show the following: (a) Given (square) matrices A and B, then AB may not equal BA, but show that det(AB) equals the det(BA). (b) Given (square) matrices A and B, assume B is invertible and show that: det !! = det () (3) [8] Given the matrix A: 3 = 3 0 8 1 3 1 6 1 3 0 2 3 9 1 4 3 9 2 6 Find a basis for column space of A and the kernel (null space) of A. (4) [8] Write the vector ! line = !. 6 as the sum of two vectors, one on the line = 2 and one on the 5 (5) [8] Define a linear transformation: : ! ! = (0) (0) Find the polynomials ! and ! in ! that span the kernel of T and describe the range of T. (6) [8] In the vector space of all real valued functions, find a basis for the subspace spanned by {cos , cos 2 , sin } (7) [10] Suppose = , and = , are bases of ! , find the change of basis matrix if = 6 4 4 3 , = , = , = 12 2 2 9 (8) [6] Suppose that A is an matrix with n real eigenvalues ! , ! , ... , ! repeated according to their multiplicity, show that trace A = ! +! + + ! . (The trace of a matrix is the sum of its diagonal elements.) (9) [10] For the given matrix A, find the eigenvalues and eigenvectors, and the eigenbasis (if it exists): 2 4 3 = 2 6 3 1 3 1 (10) [10] Find the least square solution for: 1 3 5 = 1 1 0 , 1 1 2 1 3 3 3 = 7 5 3 (11) [8] This paragraph from the text defines the term \"controllable\". Determine of the following matrix pair is controllable: 0.9 1.0 0 = 0 0.9 0 , 0 0 0.5 (12) 1 = 1 0 [8] Consider the linear transformation from ! into ! defined by: = 1 3 2 6 Do the following: (a) Determine the kernel of this transformation. (b) Graph the linear subspace defined by the kernel. (c) Determine if the vector = 1 is in the range of A. If so, what is its preimage? 2