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Unformatted text preview: (1) Suppose that the system of equations described by the augmented matrix a b c d p q is inconsistent (has no solutions). Show that the vectors ( a,b ) and ( c,d ) must be parallel. (2) Show that ( x 1 ,x 2 ,x 3 ) Span ( (2 , 1 , 7) , (3 , 2 , 6) ) if and only if 8 x 1-9 x 2-x 3 = 0. (3) Prove that the function T ( x 1 ,x 2 ) = ( x 1-2 x 2 , 4 x 2 ) is a linear transformation (4) (a) Suppose that A is an m n matrix with Ax = 0 for all x R m . Show that A = 0. (b) Suppose that A,B are m n matrices with Ax = Bx for all x R m . Show that A = B . (5) Let ~x,~ y be in R n . Show that proj ~ y ~x and ~x-proj ~ y ~x are perpendicular. (6) Let ~x R n . Show that the function f : R n R 1 = R dened by f ( ~ y ) = ~x ~ y is a linear transformation....
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