I am trying to understand the following basic proposition about invertibility: a linear map is invertible if and only if it is injective and surjective.(adsbygoogle = window.adsbygoogle || []).push({});

Now suppose ##T## is a linear map ##T:V\rightarrow W##. The book I read goes the following way in proving the proposition in the direction when the surjectivity and injectivity act as the condition. Suppose ##T## is injective and surjective and a vector ##w \in W##. Then define ##Sw## to be the unique element of V such that ##TSw = w##. Therefore ##TS## is an identity transformation in ##W##.

Now, I understand that ##Sw## is unique because ##T## is injective, but I don't know how the surjectivity contributes to guarantee that ##Sw## which satisfies ##TSw = w## does exist.

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# Proof of invertibility

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