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**HallsofIvy** When L is written as a matrix, "with respect to bases B for V, and C for W", you get the columns of A by applying L to each of the vectors in B in turn, writing the result as linear combination of vectors in C. That means, in particular, that if [tex]a_1, a_2, \cdot\cdot\cdot a_m[\math] is the first column, The $\displaystyle L(v_1)= a_1w_1+ a_2w_2+ \cdot\cdot\cdot+ a_nw_n$ where $\displaystyle v_1$ is the first vector in basis B and $\displaystyle w_1, w_2, \cdot\cdot\cdot w_n$ are the vectors in basis C. That is, in that sense, every column of A represents a vector in W.