Question

Let gamma = {t^2 - t + 1, t + 1, t^2 + 1} and beta = {t^2 + t + 4, 4t^2 - 3t + 2, 2t^2 + 3} be ordered bases for P_2(R).Find the change of coordinate matrix Q

Alternate coordinate systems
ANSWERED
asked 2020-11-01

Let \(\displaystyle\gamma={\left\lbrace{t}^{{2}}-{t}+{1},{t}+{1},{t}^{{2}}+{1}\right\rbrace}{\quad\text{and}\quad}\beta={\left\lbrace{t}^{{2}}+{t}+{4},{4}{t}^{{2}}-{3}{t}+{2},{2}{t}^{{2}}+{3}\right\rbrace}{b}{e}{\quad\text{or}\quad}{d}{e}{r}{e}{d}{b}{a}{s}{e}{s}{f}{\quad\text{or}\quad}{P}_{{2}}{\left({R}\right)}.\) Find the change of coordinate matrix Q that changes \(\beta \text{ coordinates into } \gamma-\text{ coordinates}\)

Answers (1)

2020-11-02

Let \(\displaystyle{t}^{{2}}+{t}+{4}=\gamma_{{1}}{\left({t}^{{2}}-{t}+{1}\right)}+\gamma_{{2}}{\left({t}={1}\right)}+\gamma_{{3}}{\left({t}^{{2}}+{1}\right)}\)
\(\displaystyle\Rightarrow\gamma_{{1}}+\gamma_{{3}}={1},-\gamma_{{1}}+\gamma_{{2}}={1},\gamma_{{1}}+\gamma_{{2}}+\gamma_{{3}}={4}\)
\(\displaystyle\Rightarrow{1}+\gamma_{{2}}={4}\)
\(\displaystyle\Rightarrow\gamma_{{2}}={3}\)
\(\displaystyle-\gamma_{{1}}+\gamma_{{2}}={1}\Rightarrow-\gamma_{{1}}+{3}={1}\Rightarrow\gamma_{{1}}={2}\)
\(\displaystyle\gamma_{{1}}+\gamma_{{3}}={1}\Rightarrow{2}+\gamma_{{3}}={1}\Rightarrow\gamma_{{3}}=-{1}\) Let \(\displaystyle{4}{t}^{{2}}-{3}{r}+{2}=\gamma_{{1}}{\left({t}^{{2}}-{t}+{1}\right)}+\gamma_{{2}}{\left({t}+{1}\right)}+\gamma_{{3}}{\left({t}^{{2}}+{1}\right)}\)
\(\displaystyle\Rightarrow\gamma_{{1}}+\gamma_{{3}}={4},-\gamma_{{1}}+\gamma_{{2}}=-{3},\gamma_{{1}}+\gamma_{{2}}+\gamma_{{3}}={2}\)
\(\displaystyle\Rightarrow{4}+\gamma_{{2}}={2}\)
\(\displaystyle\Rightarrow\gamma_{{2}}=-{2}\)
\(\displaystyle-\gamma_{{1}}+\gamma_{{2}}=-{3}\Rightarrow-\gamma_{{1}}-{2}=-{3}\Rightarrow\gamma_{{1}}={1}\)
\(\displaystyle\gamma_{{1}}+\gamma_{{3}}={4}\Rightarrow{1}+\gamma_{{3}}={4}\Rightarrow\gamma_{{3}}={3}\) Let \(\displaystyle{2}{t}^{{2}}+{3}=\gamma_{{1}}{\left({t}^{{2}}-{t}+{1}\right)}+\gamma_{{2}}{\left({t}+{1}\right)}+\gamma_{{3}}{\left({t}^{{2}}+{1}\right)}\)
\(\displaystyle\Rightarrow\gamma_{{1}}+\gamma_{{3}}={2},-\gamma_{{1}}+\gamma_{{2}}={0},\gamma_{{1}}+\gamma_{{2}}+\gamma_{{3}}={3}\)
\(\displaystyle\Rightarrow{2}+\gamma_{{2}}={3}\)
\(\displaystyle\Rightarrow\gamma_{{2}}={1}\)
\(\displaystyle-\gamma_{{1}}+\gamma_{{2}}={0}\Rightarrow-\gamma_{{1}}+{1}={0}{R}{i}{>}\leftrightarrow{o}{w}\gamma_{{1}}={1}\)
\(\displaystyle\gamma_{{1}}+\gamma_{{3}}={2}\Rightarrow{1}+\gamma_{{3}}={2}\Rightarrow\gamma_{{3}}={1}\)
\(\therefore Q = \left(\begin{array}{c}2 & 1 & 1 \\ 3 & -2 & 1 & \\-1 & 3 & 1\end{array}\right)\)

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