Linear Transformations Determine if the following is a linear transformation. If so, find the standard matrix of the transformation.
$T:P_1\to P_2$ such that $$T(p(t))= \int p(t) dt$$
I'd appreciate any help, an explanation of how to approach the problem would be extremely helpful. Thank you!
 A: Let's us use the standard basis for $P_1$, the vector space of polynomials of degree at most $1$. We will define the mapping
$$T(p)=\int_0^x p(t)\ dt$$
notice that we need to take a definite integral to avoid ambiguity in the constant of integration.
Let us first prove that this mapping is linear. If we let $p$ and $q$ be polynomials in $P_1$ with scalar $c$, then we have
$$T(cp + q) = \int_0^x cp(t) + q(t)\ dt = c\int_0^x p(t)\ dt + \int_0^x q(t)\ dt = cT(p) + T(q)$$
so the mapping is indeed linear.
If we feed the standard basis vectors into the mapping, we end up with
$$T(1) = \int_0^x 1\ dt = x$$
$$T(t) = \int_0^x t\ dt = \frac{x^2}{2}$$ 
We can write the matrix of the mapping $T$ with respect to the standard basis vectors of $P_1$ and $P_2$ as
$$[T] = \begin{pmatrix}0 & 0 \\ 1 & 0 \\ 0 & \frac{1}{2}\end{pmatrix}$$
A: If your definition of a linear transformation matches Wikipedia, you need to prove that $T(kx)=kT(x)$ and $T(x+y)=T(x)+T(y)$.  Both of these are usually proved properties of integrals, so go back to the proofs of these.
