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The question is "Suppose T:V--> W is a linear transformation of rank r. Prove that there exist bases B = {b1,…,bm} for V and C=[c1,…cn} for W such that T(bi)={ci, if 1<= i <= r, 0 if i > r} Describe, by explicitly giving all the entries, the matrix for T with respect to the bases B and C."

Here is my attempt: By definition of the matrix of a linear transformation (pg 289), the matrix for T relative to the bases B and C is M = [[T(b1)]C . . . [T(bi)]C . . . [T(bn)]C] So M = [[1 0], . . . , [i 0], . . . , [r 0]]

I'm not sure I am even close. Some guidance would be greatly appreciated!

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Take a basis for range of $T$ extend it to the basis of $W$. look at preimage of basis of image of $T$. prove that they are linearly independent and extend it to a basis of $V$. now check that you are through.

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