Example of a non-isomorphic linear transformation

Question

This answer makes sense to me except for the last part. Could anyone provide a simple example for me?

For example I'm thinking c0=1,c1=2,c2=2,and c3=3

So the polynomial formed would be f(t) = (t-1)(t-3) = t^2-4t+3

But how exactly does f(ci) = 0 for those? wouldn't it not work for the non-distinct numbers? Thanks!

Answer 1

Well, for your concrete example of $c_i$ we can take $$f(t) = (t - 1)(t - 2)(t - 3) = (t^2 - 3t + 2)(t - 3) = t^3 - 6t^2 - 7t - 6. $$

Then

$$ T(f) = \begin{pmatrix} f(1) \\ f(2) \\ f(2) \\ f(3) \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}. $$

The polynomial $f$ is a polynomial of degree $3$ which is non-zero but $T(f) = 0$.