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I was reading Hoffman and Kunze where I came across the following definition:

Let us call a linear transformation T non-singular if T(a) = 0 implies that a = 0, i.e., if the null space of T is {0}. Evidently, T is 1:1 iff T is non-singular.

So what I gathered was, Saying a linear transformation is non-singular is the same as saying it is 1:1. But a few pages later they define the following:

If V and W are vector spaces over the field F, any one-one linear transformation T of V onto W is called an isomorphism of V onto W. If there exists an isomorphism of V onto W, we say that V is isomorphic to W.

So don't isomorphism and non-singular mean the same thing? Why would we want to give it two names?

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# Does isomorphic imply

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