Linear transformation formatting question

Question

Let ξ,η,γ,δ ∈ F be fixed scalars.
Determine which of these maps T : F2 → F2 are linear transformations:

(i) $T(α,β)=(ξα+ηβ,γα+δβ).$
(ii) $T(α,β)=(ξα^2 +ηβ^2,γα^2 +δβ^2)$.
(iii) $T(α,β)=(ξ^2α+η^2β,γ^2α+δ^2β$).

So this is the problem I'm trying to solve, however I'm stuck on interpreting how the transformations are written.

What I have so far:
for(i):
$α$ $β$ are the two vectors in question
$α$ and $β$ themselves are just [$α,α$] and [$β$,$β$]
T($α$) is [$ξα,γα$] and $β$ is the same but with its given variables
Then to verify the first property:$F(x+y)= f(x)+f(y)$
I would rewrite it as $T(α)=(ξα,γα) T(β)=(ηβ,δβ)$
And then $T(α)+T(β)$= $(ξα+ηβ,γα+δβ)$
For the second property would something like this work: $cT(α)$=$(cξα,cγα)$=$c(ξα,γα)$
(And ii, iii are pretty much the same)
I'm guessing theres a couple mistakes somewhere. Thanks

Answer 1

For (i) we can write

$$T(\alpha,\beta)=\begin{pmatrix} ξ & η \\ γ & δ \end{pmatrix}(\alpha,\beta)^T=$$

thus it is a linear transformation.

For (iii) it is similar to (i).

For (ii) note that $$T(2α,2β)\neq 2T(α,β)$$