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Unformatted text preview: (1) Suppose that the system of equations described by the augmented matrix ±± a b c d ² ³ ³ ³ ³ ± p q ²² 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 19 x 2x 3 = 0. (3) Prove that the function T ( x 1 ,x 2 ) = ( x 12 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 ~xproj ~ y ~x are perpendicular. (6) Let ~x ∈ R n . Show that the function f : R n → R 1 = R deﬁned by f ( ~ y ) = ~x · ~ y is a linear transformation....
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 Fall '08
 Staff
 Linear Algebra, Algebra, Equations, Vectors, augmented matrix, linear transformation, parametric description

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