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Vector Spaces, Invertible Linear Operator and Change of Basis

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Let be a basis of a vector space over and let be an invertible linear operator on . Then is also a basis of
Show that ,

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Vector Spaces, Invertible Linear Operator and Change of Basis are investigated.

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Proof:
First, I show that B'={Tu_1,...,Tu_n} is also a basis of V.
We consider any a_1,...,a_n in K
If a_1*Tu_1+a_2*Tu_2+...+a_n*Tu_n = 0
Since T is a linear operator, then ...

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