## Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials

## Problem 481

Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less.

Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where

\begin{align*}

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

p_3(x)&=1+2x+8x^2, \quad p_4(x)=1+x+x^2.

\end{align*}

**(a)** Find a basis of $P_2$ among the vectors of $S$. (Explain why it is a basis of $P_2$.)

**(b)** Let $B’$ be the basis you obtained in part (a).

For each vector of $S$ which is not in $B’$, find the coordinate vector of it with respect to the basis $B’$.

(The Ohio State University, Linear Algebra Final Exam Problem)

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