Consider the function $T: P_1 \rightarrow P_1$ defined by $T(at+b) = (3a+b)t + 4a +2b$.
Show that T is a linear transformation.
Solution: From the definition of a linear transformation we must show that the function satisfy the two properties of a linear transformation(vector addition and scalar multiplication)
This is what I have:
$T((at+b)+(a_1t+b_1)) = T((a+a_1)t + (b+b_1)) = T(at+b)+T(a_1t+b_1)$
But I don't think this shows that the function holds for vector addition.