Thus every polynomial is expressible as a linear combination of the vectors in this set. But then which implies that is linearly independent. This is clearly false, hence a contradiction. Thus the vector space of polynomials is infinite dimensional.

Why are polynomials infinite dimensional?

The vector space of polynomials in x with rational coefficients. Not every vector space is given by the span of a finite number of vectors. Such a vector space is said to be of infinite dimension or infinite dimensional.

What is the dimension of polynomials?

The dimension of the vector space of polynomials in x with real coefficients having degree at most two is 3. A vector space that consists of only the zero vector has dimension zero. It can be shown that every set of linearly independent vectors in V has size at most dim(V).