082nd_tut8sol

# 082nd_tut8sol - MATH1111/2008-09/Tutorial VIII Solution 1...

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MATH1111/2008-09/Tutorial VIII Solution 1 Tutorial VIII Suggested Solution 1. Determine whether the following are linear transformations. (a) L : R 2 R 3 , L ( x ) = ( x 1 x 2 1) T . (b) L : R 2 R 3 , L ( x ) = (0 0 x 1 x 2 ) T . (c) L : C [0 . 1] C [0 , 1], L ( f ) = F where F ( x ) = R x 0 f ( t ) dt and C [0 , 1] is the vector space of all continuous functions on the interval [0 , 1]. Ans . (a) No. L (0 ) = 0 when L is a linear transformation. However the function in this case does not satisfy this property. (b) No. L ( e 1 ) = L ( e 2 ) = 0 , however, L ( e 1 + e 2 ) = (0 0 1) T 6 = L ( e 1 ) + L ( e 2 ). (c) Yes. Let f,g C [0 , 1]. Then L ( f ) = R x 0 f ( t ) dt , L ( g ) = R x 0 g ( t ) dt and L ( f + g ) = Z x 0 ( f ( t ) + g ( t ) ) dt = Z x 0 f ( t ) dt + Z x 0 g ( t ) dt = L ( f ) + L ( g ) . 2. Is it possible to construct a linear transformation fulﬁlling the following requirements? Describe the linear transformation if yes, or give an explanation if no. (a)

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082nd_tut8sol - MATH1111/2008-09/Tutorial VIII Solution 1...

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