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Suppose $v_1, \dots v_n$ spans V and $T \in L(V, W)$. Prove that the list $Tv_1, \dots , Tv_n$ spans rangeT.

I said that if $v \in V = a_1v_1 + \dots + a_nv_n$

then $T(a_1v_1 + \dots + a_nv_n) = a_1w_1 + \dots + a_nw_n$ but how do I show this spans W if I don't know that T is surjective? Thanks!

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Let $y\in \operatorname{im}T$ so there's $x=x_1 v_1+\cdots+x_n v_n\in V$ such that

$$T(x)=T(x_1 v_1+\cdots+x_n v_n)=x_1T(v_1)+\cdots+x_n T(v_n)=y$$ hence we proved that $$\operatorname{im}T\subset \operatorname{span}(T(v_1),\ldots, T(v_n))$$ and the other inclusion is clear hence the equality.

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  • $\begingroup$ how can you assume the span of the image is equal to the entire range of T?? $\endgroup$ – Soaps Jul 31 '14 at 6:17
  • $\begingroup$ By double inclusion. $\endgroup$ – user63181 Jul 31 '14 at 6:27
  • $\begingroup$ @Soaps you pick a generic element in range$T$, namely $y$, and show that this element is contained in the span of the $Tv_i$. $\endgroup$ – Daniel Valenzuela Jul 31 '14 at 7:28

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