Exam1_Solutions_Fall_2013_mod

T f a ny linear combination of vectors can alw ays be

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Unformatted text preview: A ny linear combination of vectors can alw ays be w ritten in the form Ax for a suitable m atrix A and vector x. __F_g. T he e quation x = p + tv describes a line through v parallel to p . __F_h. T he columns of a matrix A are linearly independent if the equation Ax = 0 has the trivial solution. 2 . F ind the general solutions of the systems w hose augmented m atrix is: 1 2 1 3 0 1 2 1 2 4 5 5 3 0 0 3 3 6 6 8 3 3 6 6 1 2 1 3 0 0 0 3 1 3 0 0 0 0 0 T hus x4 is free, x3 (3 1x4 ) / 3 1 x4 / 3, 0 1 2 1 3 0 1 3 0 0 3 1 3 8 3 0 0 3 1 3 3 x2 is free , and x1 (0 (2) x2 (1) x3 3x4 ) /1 1 2 x2 10 x4 / 3, 3 . Show that if Ax b and Ay b , then for any scalar , A( ( y x)) 0. If Ax b and Ay b then for any scalar , A ( y x) A( y x) ( Ay Ax) (b b) 0 0 . 4 . Suppose y ou are to solve m different linear systems of n e quations in n unknow ns. A ll of the equations have the same matrix, how ever, they just differ in right hand sides. Estimate how many a dditions are required. n3 additions and each of the m different 3 n3 2 mn 2 additions. linear...
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