Prove that If linear transformation T V to W both vector spa

Prove that If linear transformation T: V to W (both vector spaces) has kernel = {0} then T is a 1-1 linear transformation.

Solution

Since ker T ( nullity)= {0}. Let for x,y € V, T(x)=T(y). Implies T(x)-T(y) = 0

Implies x-y= 0. This implies x is additive inverse of -y and -y is additive inverse ofx. But also y is additive inverse of -y .vthetefore by uniqueness of additive inverses x= y. Thus T is one one.