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math110s-hw6

# math110s-hw6 - MATH 110 LINEAR ALGEBRA SPRING 2007/08...

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MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 6 If T : V V is a linear operator, we will write T 2 = T ◦T , T 3 = T ◦T ◦T , etc. We let O : V V and I : V V denote the zero and identity operators, ie. O ( v ) = 0 V and I ( v ) = v for all v V . 1. Let V and W be finite dimensional vector spaces over F . Let T : V W be a linear transfor- mation. (a) If V 0 is a subspace of V , show that the set T ( V 0 ) := {T ( v ) W | v V 0 } is a subspace of W . (b) If W 0 is a subspace of W , show that the set T - 1 ( W 0 ) := { v V | T ( v ) W 0 } is a subspace of V . [ Note : T - 1 ( W 0 ) is just a notation for the set defined by the rhs , T is not necessarily invertible.] (c) Deduce that T ( V ) = im( T ) and T - 1 ( { 0 W } ) = ker( T ). What are T ( { 0 V } ) and T - 1 ( W )? What about T (ker( T )) and T - 1 (im( T ))? (d) Let B W = { w 1 , . . . , w m } be a basis for im( T ). If v 1 , . . . , v m V are such that T ( v i ) = w i for i = 1 , . . . , m , show that span { v 1 , . . . , v m } ⊆ T - 1 (im( T )) but equality need not hold in general. (e) Prove that span { v 1 , . . . , v m } ⊕ ker( T ) = V and deduce the rank-nullity theorem, ie.

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math110s-hw6 - MATH 110 LINEAR ALGEBRA SPRING 2007/08...

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