Question

Why is this true?
If the linear transformation T :Rn→Rn is one-to-one
and T (u − v) = 0, then u = v.

Answer 1

Since the transformation is linear, then \(T(u-v)=T(u)-T(v)=0 \implies T(u)=T(v)\). And since it's one-to-one, that means \(u=v\).

Answer 2

A one-to-one function has the property that if \(T(v)=T(u)\), then \(v=u\).