By Otto Bretscher, Kyle Burke
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This handbook includes thoroughly worked-out options for all of the odd-numbered routines within the textual content.
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Extra info for Student Solutions Manual for Linear Algebra with Applications
Example text
F; For any 2 × 2 matrix A, the two columns of A 23. F; A reflection matrix is of the form a b 1 1 1 1 will be identical. b , where a2 +b2 = 1. Here, a2 +b2 = 1+1 = 2. −a 25. T; The product is det(A)I2 . 27. T; Note that the matrix 0 −1 1 0 represents a rotation through π/2. Thus n = 4 (or any multiple of 4) works. 29. F; If matrix A has two identical rows, then so does AB, for any matrix B. Thus AB cannot be In , so that A fails to be invertible. 31. F; Consider the matrix A that represents a rotation through the angle 2π/17.
2 1 37. 3 Since k is between 0 and 1, the tip of this vector T (x) is on the line segment connecting the tips of T (v) and T (w). 37 . x1 x1 T (e1 ) . . T (em ) 39. 2, we have T . . = . . = x1 T (e1 ) + · · · + xm xm xm T (em ). 33 Chapter 2 SSM: Linear Algebra 41. These linear transformations are of the form [y] = [a b] x1 , or y = ax1 + bx2 . The x2 graph of such a function is a plane through the origin. 2 x1 x1 43. a. T (x) = 3 · x2 = 2x1 + 3x2 + 4x3 = [2 3 4] x2 x3 4 x3 The transformation is indeed linear, with matrix [2 3 4].
Consider the linear transformation T with matrix A = [w1 T x1 x2 =A x1 x2 = [w1 w2 ] x1 x2 w2 ], that is, = x1 w 1 + x2 w 2 . The curve C is the image of the unit circle under the transformation T : if v = cos(t) is on the unit circle, then T (v) = cos(t)w1 + sin(t)w2 is on the curve sin(t) C. Therefore, C is an ellipse, by Exercise 50. 49 . 51 . 3 .. 1. rref 2 3.. 1 0 5 8.. 0 1 .. 1 0. 8 −3 = , so that .. 0 1. −5 2 45 2 3 5 8 −1 = 8 −3 . −5 2 Chapter 2 SSM: Linear Algebra . −1 − 21 1 0..