1A. MillerM340 Final ExamMay 13, 1993There is a total of 150 points. Show all work.1. (15 points) What is the determinant of (AB)T?A =1 0 0 06 −1 0 07 0 2 02 1 0 3B =2 1 1 40 −1 2 00 0 1 −30 0 0 42. (15 points) Let A be an m × n matrix and X =x1...xnan n × 1 columnvector. Prove thatAX = x1col1(A) + . . . + xncoln(A).3. (15 points)A =1 −2 00 0 20 0 −1Find an invertible matrix P and diagonal matrix D with A = P DP−1.4. (15 points) Use Cramer’s rule to solve:x − y = 22x + y = 35. (15 points) Suppose A is an invertible matrix and λ is an eigenvalue ofA.(a) Prove that λ 6= 0.(b) Prove that1λis an eigenvalue of A−1.26. (15 points) Suppose L : V → W is a linear transformation, the nullspaceof L has only zero vector in it (null(L) = {0}), and v1, . . . , vnare linearlyindependent vectors in V . Prove that L(v1), . . . , L(vn) are linearly indepen-dent.7. (20 points)A =0 −1 1 0 −10 1 −1 0 10 1 −1 1 −2(a) Find a basis for the range space of A.(b) Find a basis for the null space of A.8. (15 points) Suppose A is a 13 × 12 matrix, and B is a 12 × 13 matrix,and C = AB. Prove that C is not invertible.9. (25 points) Let R4×1have the inner product defined byhx1x2x3x4,y1y2y3y4i = x1y1+ 2x2y2+ 2x3y3+ x4y4and use this inner product in the following problem. Letv1=−1110v2=0−111v3=1111Find a vector w ∈ span{v1, v2, v3} other than the zero vector such that w isorthogonal to both v1and
View Full Document
Unlocking...