Basis of the polynomial vector space

A basis for a polynomial vector space $P=\{ p_1,p_2,\ldots,p_n \}$ is a set of vectors (polynomials in this case) that spans the space, and is linearly independent. Take for example, $$S=\{ 1,x,x^2 \}.$$ This spans the set of all polynomials ($P_2$) of the form $$ax^2+bx+c,$$ and one vector in $S$ cannot be written as a multiple of the other two. The vector space $\{ 1,x,x^2,x^2+1 \}$ on the other hand spans the space, but the 4th vector can be written as a multiple of the first and third (not linearly independent), thus it is not a basis.

The simplest possible basis is the monomial basis: $\{1,x,x^2,x^3,\ldots,x^n\}$.

Recall the definition of a basis. The key property is that some linear combination of basis vectors can represent any vector in the space.

If, instead of thinking of vectors as tuples such as $[1\ 2\ 4]$, you think of them as polynomials in and of themselves, then you see that you can make any real-valued polynomial of degree less than or equal to $n$ out of the monomial basis listed above.

You don't have to take the monomial basis. For example, you could have $\{1, x^2-4, x^3+x\}$ as a basis. But you cannot make every possible polynomial of degree $\le 3$ out of this basis!

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.