How to find T(v) given T on basis vectors in R^2

[RadiationX]

I'm having some difficulty understanding how to perform linear transformations on matrices. I understand the definition but not how to perform the operations. I'm going to give a few examples from my book:

Suppose that $$T: R^2 \longrightarrow R^2$$ is a linear transformation such that.

$$T\left(\left[\begin{array}{cc}1\\1\end{array}\right]\right)=\left( \left[\begin{array}{cc}2\\3\end{array}\right]\right)$$

And

$$T\left(\left[\begin{array}{cc}1\\-1\end{array}\right]\right)= \left(\left[\begin{array}{cc}4\\-1\end{array}\right]\right)$$


(a) Find: $$T\left(\left[\begin{array}{cc}3\\3\end{array}\right]\right)$$


Solution:


since:
$$\left(\left[\begin{array}{cc}3\\3\end{array}\right]\right) = 3\left(\left[\begin{array}{cc}1\\1\end{array}\right]\right)$$

it follows that

$$T\left(\left[\begin{array}{cc}3\\3\end{array}\right]\right) = T3\left(\left[\begin{array}{cc}1\\1\end{array}\right]\right) = 3T\left(\left[\begin{array}{cc}1\\1\end{array}\right]\right) = 3\left(\left[\begin{array}{cc}2\\3\end{array}\right]\right) = \left(\left[\begin{array}{cc}6\\9\end{array}\right]\right)$$


How do they come to this solution?

 

[Mmmm]

it follows from the definition of a linear transformation:

If T is a linear transformation, then
$$T(av_1 +bv_2) = aT(v_1)+bT(v_2)$$

a and b are constants and $v_1$ and $v_2$ are vectors.