What is the Relationship Between Injectivity and Surjectivity in Linear Algebra?

[kylem]

I'm having a lot of trouble figuring out how to do this proof:

Given real numbers a, b, c, d, let f: real^2 ----> real^2 be defined by f(x, y) = (ax + by, cx + dy). Prove that f is injective if and only if f is surjective.

If anybody could help, that would be great.

 

[Office_Shredder]

Do you know what the rank-nullity theorem is?

 

[kylem]

Do you know what the rank-nullity theorem is?

A quick google search reveals that it appears to be dealing with some linear algebra I don't quite understand...

My book gives me the following hint for this proof: Consider two cases, depending on whether ad - bc = 0.

Not sure if that helps, I'm pretty confused here.