Linear Algebra: How to show this transformation is linear? 
Let $B$ be the standard basis for $P_{3}$, and let $T: P_{3} \rightarrow P_{4}$ be defined by $T(x^k) = \int_0^k t^k \,dt$.
Show that $T$ is a linear transformation.

I'm aware of the conditions to show a linear transformation, but the variables have left me a little confused in this specific problem. Thanks!

This is an odd question. There is an "obvious" question that's hiding behind this question, but the original question is answerable. 
We first address the original question, and then give the question this question probably meant to be.
Original Question
Given only the question as stated, how do we define $T(1+2x)$? We know how to define $T(1)$ and $T(x)$, but sums or products are ambiguous. It is stated that $B$ is the standard basis for $P_3$, but no additional reference is given to $B$.
It seems most obvious that one is defining $T$ on the basis elements of $B$ and "extending by linearity." That is, it seems most obvious that one should define $T(1 + 2x) = T(1) + 2 T(x)$. Defining a map on the basis elements in this way is tautologically a linear map, as one uses linearity to define the map. Using no properties of $T$ aside from the fact that it's defined on the basis elements, this interpretation of $T: P_3 
\longrightarrow P_4$ leads to a (trivially) linear map.
In this definition, what is $T(1 + 2x)$? We have
$$ T(1 + 2x) = T(1) + 2 T(x) = \int_0^0 t^0 dt + 2 \int_0^1 t^1 dt = 2.$$
For another check, to make sure we understand, we compute
$$ T(1 + 2x + 3x^2) = T(1) + 2 T(x) + 3 T(x^2) = 0 + 2 + 3 \int_0^2 t^2 dt = 2 + 3 \cdot \frac{3^3}{3} = 29.$$
Thus this is a well-defined linear map. It may be interesting to investigate this linear map. But being asked to prove it is linear is a somewhat malformed question.
The "Obvious" Question behind this question
Those familiar with the subject material will be inclined to think the question meant to say the following.

Let $f(t) \in P_3$ be a degree $3$ polynomial over $\mathbb{R}$. Define
  $T: P_3 \longrightarrow P_4$ by
  $$ T(f) := \int_0^x f(t) dt.$$
  Show that $T$ is a linear map.

Natural follow-ups include determining the image, kernel, and the orthogonal space to the image. This is a classical problem.

You might benefit from understanding why the original question is tautological or malformed, and then from tackling the modified problem above to make sure you understand how linear maps from spaces of functions to other spaces of functions work.
