Thanks for such a quick response. I must've edited my post 5 times while you answered, so my original post is a bit different than what you read.

I can see. Your technique is much better than my book's technique.

you did:

and got

as in, you did both sides at once to prove they are unequal.

The way my book answers the questions is by starting with the first step: T(a+b), doing all the work from that, and finally coming to the conclusion that it equals T(a) + T(b). For example, the way I tried to answer was like this:

T(a+b) = T(\alpha_1 + \beta_1, \alpha_2 + \beta_2)=

=((\alpha_1 + \beta_1)^2, (\alpha_1 + \beta_1)(\alpha_2 + \beta_2))=

= (\alpha_1^2 + 2\alpha_1\beta_1 + \beta_1^2), (\alpha_1\alpha_2 + \alpha_1\beta_2 + \beta_1\alpha_2 + \beta_1\beta_2)

and now I am having trouble proving or disproving if this equals T(a) + T(b).

It can, but it would

**not **be a

**linear **transformation

Correct?