Can a list of $m+1$ polynomials with degree at most $m$ be linearly independent if at least two elements of the list have the same degree?

Question

Given the vector space of polynomials with degree at most $m$, can a list of $m + 1$ polynomials in this vector space be linearly dependent if at least two of the polynomials have the same degree?

It seems the converse is proven here but I can't seem to be able to derive the question here, even though it seems probably true.

Answer 1

Yes. For example, consider the case $m = 1$ and the set $$\{x, x + 1\} .$$