1. Consider the following two ordered bases: Bu = {(5), (3) B2 = { 1-2) /1 1-2 ) B = 11-5)' ( 2 ) (a) Find the transition matrix from B1 to B2. (b) Use your transition matrix from part (a) to compute the Bi-coordinate vector for [(3 4)T]12
5. (12 pts) Let A= 4 -1 2 -1 3 -3 2 0 2 1 Find A-? using the formula A-1 adj(A). det(A)
1 3-1 2 Let A=1 12 1 1 18 24 65 (a) Find the rank of the matrix (b) Is the matrix A invertible? Then find its inverse
3. Let La A = 1 - 2 5 -3 2 5 0 -12-2 . L (a) (8 points) It turns out that the matrix equation Ax = b is consistent only for a special type of vector b where bi, b2, and b3 satisfy a certain equation. Find this equation. (b) (8 points) The set of all vectors satisfying the equation found in part (a) equals Span {W1, w2} Find wį and w2.
2 3 12 3 37 1. Let A - 10 15 40 7 1131 2 3 7 2 2 and B 1-2 -3 8 3 171 echelon form of A. (Assume this!) (a) (2 pt) What is the value of rank(A)? 110057 100 100 000121The B is the reduced to loooool (b) (2 pt) What is the value of nullity(AT)? (Read carefully (C) (3 pt) Find a basis for col(A). Circle your final answer. (d) (3 pt) Find a basis...
Answer part c (ii) (b) Let 12 -1 2 0 -3 0 om = A -5/ Compute the spectral radius of A.- a system of linear equations (c) Suppose a certain iterative scheme used to solve a system of lines is an invertih an invertible matrix Ax = b is given by QxK+1 = (Q - A) bu oxK+1 = (0 - AX" + b, where Prove that (1) (ii) exll s ||1 - Q - A||llex-1|| lexll s ||1...
3x + 4 for x 2 12. Let (x) 2-x for -1 <x51 . Find f(1/3) and (3/2). Sketch the graph of the -3x for x S-1 function. Determine the domain and range. (2,2,5, 3, and 3 points)
Let A = (-1 -1 2 3 1 ) 1 Let B= 2 3 3 2 -1 0 1 2 Find AB O 2 1 3 2 3 1 9 10 10 6 -40 O 1 4 1 2 -1 3 Can not multiply
3. Let B ERnxn be a symmetrie and P.D. matrix. Show that l s (o Bu) (B-v) for any nonzero v E R", and that the equality holds if and only if v is an eigenvector of B. (Hnt: note that llt -W/2t, B-1/2v), and use the Cauchy-Schwarz inequality.) 4. Let (ak) be a real sequence such that for each k, either akil > ak or akt? where, is a constant independent of k. Show that a 2 min(ai, T)...
Practice 1: Let 1 -2 3 A 4 12 3 2 1 411,E 15 4-3 -2 3 D3 2] ] , F If possible, compute each of the following 1. EB + FA 2. A(B + D) and AB + AD