quiz6-solns - = 0, we must have a = 0. The original...

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MATH 110 - SOLUTION TO QUIZ 6 LECTURE 1, SUMMER 2009 GSI: SANTIAGO CA ˜ NEZ Suppose T is an operator on V , m is a positive integer, and v V is such that T m - 1 v 6 = 0 but T m v = 0. Prove that ( v,Tv,. ..,T m - 1 v ) is linearly independent. Proof. Suppose that a 0 ,...,a m - 1 are scalars such that a 0 v + a 1 Tv + ··· + a m - 1 T m - 1 v = 0 . Since T m v = 0, applying T m - 1 to both sides of the above equation gives a 0 T m - 1 v = 0 . Thus since T m - 1 v 6
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Unformatted text preview: = 0, we must have a = 0. The original equation we had is now a 1 Tv + + a m-1 T m-1 v = 0 . Applying T m-2 to both sides will show that a 1 = 0, and continuing in this manner shows that all a i are zero. Hence ( v,Tv,. ..,T m-1 v ) is linearly independent. Date : August 6, 2009....
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