## Vector Space of Polynomials and Coordinate Vectors

## Problem 157

Let $P_2$ be the vector space of all polynomials of degree two or less.

Consider the subset in $P_2$

\[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\]
where

\begin{align*}

&p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\

&p_3(x)=2x^2, &p_4(x)=2x^2+x+1.

\end{align*}

**(a)** Use the basis $B=\{1, x, x^2\}$ of $P_2$, give the coordinate vectors of the vectors in $Q$.

**(b)** Find a basis of the span $\Span(Q)$ consisting of vectors in $Q$.

**(c)** For each vector in $Q$ which is not a basis vector you obtained in (b), express the vector as a linear combination of basis vectors.