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Mathematics 220 in-class exercises Tuesday, July 22, 2014 1. Suppose right 2 1 A= 3 4 that the matrix A on the left row-reduces to the matrix B to the 1 3 2 −1 3 −1 4 9 1 4 1 −2 1 9 1 18 1 0 B= 0 0 =⇒ reduces to 0 1 0 0 2 −1 0 0 0 0 0 0 1 0 0 1 Denote the columns of A by ~a1 , ~a2 , . . ., ~a5 , and the columns of B by ~b1 , ~b2 , . . ., ~b5 . (a) True or False: The solution set of A~x = ~0 is identical to the solution set of B~x = ~0. (b) Is ~a3 ∈ span{~a1 , ~a2 } ? (c) Is span{~a1 , ~a2 } = span{~b1 , ~b2 } ? (d) Do the columns of A form a linearly independent set? (e) Is the set {~a3 , ~a4 , ~a5 } a linearly independent set? If linear transformation T (~x) = A~x, (f) What is the domain of T ? the codomain of T ? (g) Is the linear transformation T (~x) = A~x onto its codomain? one-to-one ? (h) What is the range of T ? (i) If T (~x) = B~x, is the range of S = range of T ? 2. Suppose that A is a 3 × 3 matrix and a det(A) = r x Find the values of (a) det(3A) (b) det(A3 ) that b s y (c) det((2A)−1 ) c t z =6 (d) det(AT A) (e) det(P AP −1 ) (in part (e) assume that P is an invertible 3 × 3 matrix) a − 2x x (f) r b − 2y y s c − 2z z t a 0 (g) 2r + 3x x b 0 2s + 3y y 1 c 2 0 9 2t + 3z 14 z (h) Find the value of the matrix entry (A−1 )2,3 (i) If S is the unit ball {(x1 , x2 , x3 ) ∈ R3 : x2 + y 2 + z 2 ≤ 1}, find the volume of the image of the unit ball under multiplication by the matrix A.