# Defining an isomorphism....

• Feb 18th 2010, 07:09 AM
zhupolongjoe
Defining an isomorphism....
Prove that the field R of real numbers is isomorphic to the ring of all 2x2 matrics of the form (0 0; 0 a) with a a real number.

It says to let f(a)=(0 0; 0 a), but I don't know how to show that this is one to one and onto.
Thanks.
• Feb 18th 2010, 07:15 AM
Swlabr
Quote:

Originally Posted by zhupolongjoe
Prove that the field R of real numbers is isomorphic to the ring of all 2x2 matrics of the form (0 0; 0 a) with a a real number.

It says to let f(a)=(0 0; 0 a), but I don't know how to show that this is one to one and onto.
Thanks.

Are you sure you don't mean matrices of the form

$\left( \begin{array}{cc}
a & 0 \\
0 & a \end{array}
\right)$
?

As the set you gave has no identity.

Think about it for a bit - what does surjective mean? take an arbitrary matrix of this form. Is this mapped onto by something? What if two matrices were mapped to by the same element, what would this mean with respect to the elements which were mapping to it?

Are you confident with the definitons of surjectivity and injectivity?
• Feb 18th 2010, 07:18 AM
zhupolongjoe
No, the book says matrices of the form (0 0; 0 a). Doesn't this have identity
(0 0; 0 1)? Identity doesn't have to be (1 0; 0 1)
• Feb 18th 2010, 07:19 AM
Swlabr
Quote:

Originally Posted by zhupolongjoe
No, the book says matrices of the form (0 0; 0 a). Doesn't this have identity
(0 0; 0 1)? Identity doesn't have to be (1 0; 0 1)

Yes, you're right. One day I will learn to think before I type.
• Feb 18th 2010, 07:22 AM
clic-clac
When are two matrices equal? I guess answering this question will instantaneously make your map become one-to-one and onto.
• Feb 18th 2010, 07:48 AM
zhupolongjoe
Okay, I think I got it actually. Maybe I was just making this problem harder than it really is.