Given real numbers a,b,c,d, let, f : R(squared) go to R(squared) be defined by f(x, y) = (ax+by, cx+dy)?

How to prove that f is injective if and only if (iff) f is surjective.

Given real numbers a,b,c,d, let, f : R(squared) go to R(squared) be defined by f(x, y) = (ax+by, cx+dy)?
suppose that f is injective,


to prove: f is surjective





if f is injective then


f(1,0) = (a,c)


and


f(0,1) = ( b,d) are linearly independent vectors.


but since the dimension of R^2 is 2, then these vectors are a basis, and so


any vector (z,w) can be written as a linear combination of (a,c) and (b,d)


(z,w) = A(a,c) + B(b,d)


=( Aa + Bb, Ac+Bd)


take (x,y) = (A,B)


then f(x,y) = ( Aa + Bb, Ac+Bd) = (z,w)


thefore f is surjective.





If f is surjective, then any vector (z,w) = f(x,y) for some (x,y).