Prove the theorem Let T Rn rightarrow Rm be a linear transfo

Prove the theorem: Let T: R^n rightarrow 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.

Solution

LEt, T be one to one

Let, T(x)=T(0)=0

Since , T is one to one x=0

HEnce, T(x)=0 only has trivial solution

LEt, T(x)=0 only have trivial solution

Let, T(u)=T(v) for some u,v

T(u)-T(v)=0

T(u-v)=0

Hence, u-v=0

ie u=v

Hence, T is one to one