# Give a full correct answer for given question

Give a full correct answer for given question 1- Let W be the set of all polynomials $a+bt+c{t}^{2}\in {P}_{2}$ such that $a+b+c=0$ Show that W is a subspace of ${P}_{2},$ find a basis for W, and then find dim(W) 2 - Find two different bases of ${R}^{2}$ so that the coordinates of $b=\left[\begin{array}{c}5\\ 3\end{array}\right]$ are both (2,1) in the coordinate system defined by these two bases

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joshyoung05M

Consider $W=\left\{a+bt+c{t}^{2};a+b+c=0\right\}$ Yes.W is a subspace of ${P}_{2}.$ Since $0ϵW.$ If $x+yt+z{t}^{2}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}l+mt+n{t}^{2}$ are elements of ${P}_{2},$ then $\left(x+l\right)+\left(y+m\right)t+\left(z+n\right){t}^{2}ϵW.$ So w is closed under vector addtion. Let $rϵ\mathbb{R}$ and $a+bt+c{t}^{2}ϵW$ then $r\left(a+bt+ct2\right)=ra+rbt+rc{t}^{2}ϵW.$ Therefore w is closed under scalar multiplication . Therefore W is a subspace of ${P}_{2}.$