1. ## Span vectors

In an example from the book, the polynomials $x^2+3x-2 \ , \ 2x^2+5x-3 \ , \ -x^2-4x+4$ generate $P_{2}(R)$ since they are all in the set and each polynomial in $P_{2}(R)$ is a linear combination of these three.

What I don't understand is the what is $P_{2}(R)$ and how do you know that all poly in the set are linear combination of these three?

thank you.

In an example from the book, the polynomials $x^2+3x-2 \ , \ 2x^2+5x-3 \ , \ -x^2-4x+4$ generate $P_{2}(R)$ since they are all in the set and each polynomial in $P_{2}(R)$ is a linear combination of these three.
What I don't understand is the what is $P_{2}(R)$ and how do you know that all poly in the set are linear combination of these three?
$P_2(\mathbb{R})$ is the set of all polynomial functions up to degree $2$. We can think of a polynomial $ax^2+bx+c$ as $\left[ \begin{array}{c}a\\b\\c\end{array} \right]$. What you want to prove that that: $\left[ \begin{array}{c}1\\3\\-2\end{array} \right] , \left[ \begin{array}{c}2\\5\\-3\end{array} \right], \left[ \begin{array}{c}-1\\-4\\4\end{array}\right]$ can be used to express any vector as linear combinations of those three vectors.