Download Instructor's Solutions Manual for Linear Algebra with by Otto and Bretscher and Kyle Burke PDF

By Otto and Bretscher and Kyle Burke

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Example text

Vm ], B = [w1 . . wm ], and x =  . . . xm   x1 Then (A + B)x = [v1 + w1 . . vm + wm ]  . .  = x1 (v1 + w1 ) + · · · + xm (vm + wm ) and xm     x1 x1 Ax + Bx = [v1 . . vm ]  . .  + [w1 . . wm ]  . .  = x1 v1 + · · · + xm vm + x1 w1 + · · · + xm wm . xm xm The two results agree, as claimed. 15 we draw a typical vector in this plane. 55 We are looking for constants a and b such that a  2  + b  5  =  8 . 9 6 3     7 a + 4b = 7 The resulting system  2a + 5b = 8  has the unique solution a = −1, b = 2, so that  8  is indeed a linear 9 3a + 6b = 9     1 4 combination of the vector  2  and  5 .

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].  v1 b If v =  v2 , then T is linear with matrix [v1 v2 v3 ], as in part (a). v3           a a x1 x1 x1 c Let [a b c] be the matrix of T . Then T  x2  = [a b c]  x2  = ax1 + bx2 + cx3 =  b  ·  x2 , so that v =  b  c c x3 x3 x3 does the job. 44 T  x2  =  v2  ×  x2  =  v3 x1 − v1 x3  =  v3 −v2 v1 x2 − v2 x1 x3 v3 x3  0 −v3 v2  v3 0 −v1 .

V3           a a x1 x1 x1 c Let [a b c] be the matrix of T . Then T  x2  = [a b c]  x2  = ax1 + bx2 + cx3 =  b  ·  x2 , so that v =  b  c c x3 x3 x3 does the job. 44 T  x2  =  v2  ×  x2  =  v3 x1 − v1 x3  =  v3 −v2 v1 x2 − v2 x1 x3 v3 x3  0 −v3 v2  v3 0 −v1 . 3. Part a holds, since L(T (v + w)) = L(T (v) + T (w)) = L(T (v)) + L(T (w)), and part b also works, because L(T (kv)) = L(kT (v)) = kL(T (v)). 47 Write w as a linear combination of v1 and v2 : w = c1 v1 + c2 v2 .

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