If we define a linear map $T: V \to W$ such that $(v_1,...,v_n)$ is a basis of $V$ and we have that:

$$T(a_1v_1 + ...+a_nv_n) = a_1w_1+....+a_nw_n$$ where $a_1,...,a_n \in \mathbb{F}$. Why is it the case that we must have $Tv_j = w_j$ for $j=1,...,n$?



Put $a_i=0$ for $i\neq j$ and $a_j=1$. Then the formula you give for $T$ gives $T(v_j)=w_j$.

  • $\begingroup$ I see, so you are saying that the way I should view the transformation is that I can manipulate the coefficients however I like and that it must hold true for every iteration and permutation of the coefficients? $\endgroup$ – user123276 Feb 13 '14 at 14:08
  • $\begingroup$ Yes that's exactly how you should think about $T$. $\endgroup$ – Brian Fitzpatrick Feb 14 '14 at 0:58

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