T: R^3 ----> W , where W is the vector space of all symmetric 2x2 matrices

T [ a b c ] =

[ a+b+c b-2c ]

[ b-2c a-c ]

I'm not sure how to show it is 1 to 1. After being transformed, I see that a = c and b = 2c and 2a = -b.

But I think I know it is not "onto" because T of R^3 cannot possibly cover all of the range of M22. A side-question on a little notation problem, is the space of 2x2 matrices M22 ?