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Let $V$ and $W$ be finite dimensional vector spaces over a field $F$. Let $T:V\to W$ be a linear transformation. Suppose that $T$ is one-to-one. Show that there is a linear transformation $L:W\to V$ such that $LT=1_V$.

The second question is Let $T:V\to W$ and $L:W\to V$ be a linear transformation. Show:

  1. $T$ is injective if $LT=1_V$ and
  2. $T$ is surjective if $TL=1_W$

I know that if 1 and 2 are true, that it is a isomophism, no idea how to prove it though.

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Is this homework? What have you tried? – Matt Mar 9 '13 at 2:56
up vote 1 down vote accepted

Hint: If $T$ is one-to-one then a basis $\{v_i\}$ of $V$ gets sent to a linearly independent set $\{T(v_i)\}$ of $W$. Extend this to a basis of $W$ and define the mapping $W \to V$ by choosing where to map each element of this basis.

For (1) assume $x \in V$ is in the kernel of $T$ and then apply $LT$.

For (2) take $w \in W$, where does $L(w)$ map too?

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