Polynomial Vector Spaces

• Once we realized how powerful vector spaces are, we started using them for everything.

Here, we define $P_n(\mathbb{R})$ to be the set of all real polynomials of degree at most $n$.

$$P_n(\mathbb{R})=\{a_0+a_{1}x+...+a_n{x}^n|a_0,a_1,...a_n\in \mathbb{R}\}$$

• This gives us multiple variants like $P_1,P_2$ etc.
• We create a vector space out of this set by defining equality, addition and scalar multiplication (it’s obv, we basically dance with coefficients of the respective terms).

This makes $P_n(\mathbb{R})$ a vector space for nonnegative integers $n$.

• Since $P_n(\mathbb{R})$ is a vector space, we can define subspaces and find their bases and play with that kinda stuff.

Standard Basis

$$B=\{1,x,...,x^n\}$$