## anonymous 5 years ago $T:X \rightarrow Y$ be a linear operator and $dim X = dim Y = n < \infty$ Show that range $R(T) = Y$ if and only if $T^{-1}$ exist

1. mathteacher1729

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.

2. anonymous

yes, this one seems too easy and not very detailed