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I am just not satisfied with the following theorem (I don't know it's name):

Let T:R^n -> R^m be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution.

The "proof" involves saying that if T is not one-to-one, then there are two different vectors U and V such that T(U)=T(V)= some vector B. And since T is linear it follows that T(U-V) = T(U)-T(V) = B - B = 0. It then concludes by saying "hence there are nontrivial solutions to T(X)= 0. So, either the two conditions in the theorem are both true or they are both false."

I just don't see how that proved the theorem in any way, perhaps because I don't fully understand which two conditions it is talking about.

Could anyone help me here? Thank you.

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# Not Satisfied with Linear Algebra Theorem

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