Rank of a linear transformation.

Question

Let $V$ be the vector space of all continuous function from $[0,1]$ to $\mathbb{R}$ on the field $\mathbb{R}$. What is the rank of the linear transformation $T:V\rightarrow V $ which was defined as below? $$T(f(x))=\int_{0}^{1}(3x^3y-5x^4y^2)f(y)dy$$

Answer 1

$T(f(x)) = $$\displaystyle\int_{0}^{1}(3x^3y-5x^4y^2)f(y)dy\\ \displaystyle3x^3\int_{0}^{1}yf(y)\ dy-5x^4\int_0^1 y^2 f(y)\ dy$

$\int_{0}^{1}yf(y)\ dy$ is a constant as is $\int_0^1 y^2 f(y)\ dy$

So what does that say about $T(f(x))$?

Answer 2

Hint: $T(f)$ is a actually a polynomial...