Basis of the polynomial vector space I don't understand how to find a basis for a polynomial vector space.
Can someone help me with an example?

For an example take the vector subspace of $\mathbb{R}[x]$ of all polynomials with real coefficients of degree at most 3.  The elements in there will all look like $a_0+a_1x+a_2x^2+a_3x^3$, where the coefficients $a_i \in \mathbb{R}$ can be thought of as coordinates with respect to the set of vectors $(1,x,x^2,x^3)$.  You can check that these are linearly independent and span the space, so that you have a 4 dimensional vector space over $\mathbb{R}$.  The distinct powers of $x$ act as independent placeholders with their coefficients being coordinates.
