Polynomials are linearly independent iff rows of $A$ are linearly independent where $P(x)=A[x^k]$

Question

Polynomials are linearly independent iff rows of $A$ are linearly independent where $P(x)=A[x^k]_{k=1,\cdots,n}$. I know how to prove this but is there higher level intuition of why this is true? i.e. polynomials are no different from tuples/vectors in $R^n$