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Unformatted text preview: space over the eld of real numbers (with usual operations). Find a function from V into V that is a linear transformation on V , but that is not a linear transformation on C 1 , i.e., that is not complex linear. Exercise 13. Let V be a vector space and T a linear transformation from V into V . Prove that the following two statements about V are equivalent. (a) The intersection of the range of T and the null space of T is the zero subspace of V . 1 (b) If T ( T ) = 0, then T = 0. Bonus exercise 12. Let V be an n-dimensional vector space over the Feld F and let T be a linear transformation from V into V such that the range and null spaces of T are identical. Prove that T is even. (Can you give an example of such a linear transformation T ?) 2...
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- Spring '08