Dimension of null space of linear map? Suppose V is the real vector space of real-valued functions with a derivative. What is the dimension of the null space of the linear map
$$Tf = x\frac{df}{dx} - 4f\;\;?$$
The basis I found for the null space is $\{x^4\}$, which gives dimension 1, but the hint for problem was that the dimension is not 1. Why is that so?

We have a simple linear homogeneneous differential equation of order $\,1\,$ to solve
$$Tf=0\Longrightarrow x\frac{df}{dx}=4f\Longrightarrow \int\frac{df}{f}=4\int\frac{dx}{x}\Longrightarrow$$
$$\log|f|=4\log|x|+C\Longrightarrow f=kx^4\,\,,\,k=\,\text{a constant}$$
and yes: you're answer is correct.
Whose hint was that  $\,\dim\ker T\neq 1\,$ ?
