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When I have a linear operator $T:V\rightarrow W$ and images for some ordered basis $B=(b_1,\dots,b_n)$ of $V$, what do I have exactly if I put those images in a matrix? Is it by default:

$[T]^B_E = \begin{bmatrix} T(b_1) & \dots & T(b_n) \end{bmatrix}^B_E$

where the output is according to the standard basis of $W$? What are those images if I don't take their coordinate vectors?

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    @PatrickDaSilva What I'm talking about is a method for solving problems of this type: http://math.stackexchange.com/questions/183077/deducing-formula-for-a-linear-transformation2012-11-08

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We can define '$n$ dimension vectors' over any set $S$, as the set of all $n$-tuples formed of $S$. $S^n:=\{[ s_1,...,s_n ]\mid s_i\in S\}$ Then, what you are asking, $[T(b_1),...,T(b_n)]$ is none other than a '(row) vector over $W$', just an $n$-tuple of elements of $W$. It does make sense. Then, expanding each $T(b_i)$ to coordinate (column-)vectors by the given basis $E$ of $W$, would give the ordinary matrix $[T]^B_E$.