Simple Abstract Proof, with Matrices

[kuahji]

Prove that the field R of real numbers is isomorphic to the ring of all 2 X 2 matrices of the form (0,0)(0,a), with a as an element of R. (Hint: Consider the function f given by f(a)=(0,0)(0,a).)

I have no problem showing that it is a homomorphism & that it's injective. My question arrises for showing that it's surjective.

Can't I just choose a matrix like (1,2),(3,4), & clearly that aint happening b/c you could never find f(r)=(1,2),(3,4). We did one in class like that, & that is what the professor stated, but here in the book it says to prove it. So now I'm wondering...

 

[Dick]

It doesn't say R is isomorphic to the ring of ALL 2x2 matrices. It says it's isomorphic to the ring of all 2x2 matrices OF THE FORM (0,0),(0,a). (1,2)(3,4) isn't of that form.