Give a full correct answer for given question

Give a full correct answer for given question

Question
Alternate coordinate systems
asked 2020-11-01

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

Answers (1)

2020-11-02

Consider \(W = \left \{ a + bt + ct^{2}; a + b + c = 0 \right \}\) Yes.W is a subspace of \(\displaystyle{P}_{{{2}}}.\) Since \(\displaystyle{0}\epsilon{W}.\) If \(\displaystyle{x}+{y}{t}+{z}{t}^{{{2}}}{\quad\text{and}\quad}{l}+{m}{t}+{n}{t}^{{{2}}}\) are elements of \(\displaystyle{P}_{{{2}}},\) then \(\displaystyle{\left({x}+{l}\right)}+{\left({y}+{m}\right)}{t}+{\left({z}+{n}\right)}{t}^{{2}}\epsilon{W}.\) So w is closed under vector addtion. Let \(\displaystyle{r}\epsilon{\mathbb{{{R}}}}\) and \(\displaystyle{a}+{b}{t}+{c}{t}^{{2}}\epsilon{W}\) then \(\displaystyle{r}{\left({a}+{b}{t}+{c}{t}{2}\right)}={r}{a}+{r}{b}{t}+{r}{c}{t}^{{2}}\epsilon{W}.\) Therefore w is closed under scalar multiplication . Therefore W is a subspace of \(\displaystyle{P}_{{2}}.\)

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