Deducing formula for a linear transformation The question I'm answering is as follows:
Let $ T: \mathbb{R}^2  \rightarrow \mathbb{R}^2$ be a linear transformation such that $ T(1,1) = (2,1) $ and $ T(0,2) = (2,8) $. Find a formula for $ T(a,b) $ where $ (a,b) \in \mathbb{R}^2 $.
Earlier we proved that $\{(1,1), (0,2)\}$ spans $\mathbb{R}^2$. I used this when trying to find a formula for $T$. My working is:
$T(a(1,1) + b(0,2)) = aT(1,1) + bT(0,2) $ 
Because $T$ is linear. Thus:
$ T(a(1,1) + b(0,2)) = a(2,1) + b(2,8) = T(a,b)$
Is this correct? It seems a bit too easy and so I'm wondering if I missed anything.

We have 
$$(1,0)=(1,1)-\frac{1}{2}(0,2)\qquad\text{and} \qquad(0,1)=\frac{1}{2}(0,2).\tag{$1$}$$ 
Note that $(a,b)=a(1,0)+b(0,1)$. So 
$$T(a,b)=aT(1,0)+bT(0,1).$$
Now use the values of $T(1,1)$ and $T(0,2)$, and Equations $(1)$, to find $T(1,0)$ and $T(0,1)$, and simplify a bit.
