Question: 4. Let V be a vector space with two bases B and C. Let T : V > V be a linear transformation with B

4. Let V be a vector space with two bases B and C. Let T : V > V be a linear transformation with B as its Bmatrix and C as its C-matriX. Prove or disprove each of the following statements. (a) dim(ker(B)) =dim(ker(C)). (b) ker(B) =ker(C). (c) The sum of all entries in B is equal to the sum of all entries in C. (d) The trace (sum of the diagonal entries) of B is equal to the trace of C. Hint: It may be useful to remember the fact previously proueu on H W #3 that tracef AB ):trace( BA ) for any two n X n matrices A and B. (e) If there exists a vector 171 such that 3171 = 19171 for some scalar k E R, then there exists a vector 172 such that 0172 = 16172 for the same value of k

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