Non-singular Operator

Question

Let $V$ be the space of polynomial functions over $\mathbb{R}$

Can we give an example of a linear operator $T$ on $V$ such that $T$ is non-singular and non-invertible?

Answer 1

Let $T$ be the transformation on $R[X]$ given by

$$T\left(\sum_{i=0}^n c_i x^{i}\right) = \sum_{i=0}^n \frac{c_i}{1+i}x^{1+i}.$$

Then $T$ is non-singular (because its kernel is $0$) and non-invertible (because $1$ has no pre-image).

N.B.: $R[X]$ is infinite dimensional.