Find a linear transformation If T is a linear transformation and $$T\pmatrix{1 \\ 3} = 4$$ and  $$T\pmatrix{2 \\ -1} = -3$$
find $$T\pmatrix{7 \\ 0}$$
I think that the answer is 5 but I am not sure. 

Primes denote the transpose:
$$T((7,0)')=T(1\cdot (1,3)'+3\cdot (2,-1)')$$
$$=1\cdot T((1,3)')+3 \cdot T((2,-1)')=1\cdot 4+3\cdot (-3)=-5$$
In order to find the proper linear combination solve $(7,0)'=a(1,3)'+b(2,-1)'$, to get $a=1$ and $b=3$. At the second equality I used the linearity of $T$.
