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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 Spring '08
 PAVLOVIC
 Gaussian Elimination, Matrices, Vector Space, ith Partial Pivoting

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