\[T:X \rightarrow Y \] be a linear operator and \[dim X = dim Y = n < \infty \] Show that range \[R(T) = Y\] if and only if \[T^{-1}\] exist

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\[T:X \rightarrow Y \] be a linear operator and \[dim X = dim Y = n < \infty \] Show that range \[R(T) = Y\] if and only if \[T^{-1}\] exist

Mathematics
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You could go the linear algebra route (which you may not be allowed to do) and say something like: "since it's linear, the transformation can be represented by a square n x n matrix which, after row reduction to echelon form has a pivot in every row and every column thereby making the linear transformation one-to-one and onto." This seems too easy though... :-p Your prof might want you to work more directly from abstract set-point topological definitions.
yes, this one seems too easy and not very detailed

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