Find the coordinate vector of w relative to the basis

sklicatias 2021-11-16 Answered

Find the coordinate vector of w relative to the basis \(S = \{ u_1, u_2 \}\ for\ \mathbb{R^2}\ u_1 = (1, -1), u_2 = (1, 1) ; w = (1, 0)\)

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Answered 2021-11-17 Author has 1350 answers

Consider the basis \(\displaystyle{S}={\left\lbrace{u}_{{1}},{u}_{{2}}\right\rbrace}\ {f}{\quad\text{or}\quad}\ {\mathbb{{{R}^{{2}}}}}\) where \(u_1 = (1, -1), u_2 = (1, 1) ; w = (1, 0)\)
In order to find the coordinate vector of w relative to the basis S we must find scalars a and b such that \(\displaystyle{a}{v}_{{1}}+{b}{v}_{{2}}={w}\)
Since \(\displaystyle{a}{v}_{{1}}+{b}{v}_{{2}}={a}{\left({1},-{1}\right)}+{b}{\left({1},{1}\right)}={\left({a},-{a}\right)}+{\left({b},{b}\right)}={\left({a}+{b},-{a}+{b}\right)}\) and \(\displaystyle{w}={\left({1},{0}\right)}\) we must find scalars a and b such that \(\displaystyle{\left({a}+{b},-{a}+{b}\right)}={\left({1},{0}\right)}\) which leads us to the linear system
\(\displaystyle{a}+{b}={1}\)
\(\displaystyle-{a}+{b}={0}\)
\(\displaystyle{a}+{b}={1}\) and \(\displaystyle-{a}+{b}={0}\) gives us that \(\displaystyle{a}+{b}-{a}+{b}={1}+{0}\) which gives us that \(\displaystyle{2}{b}={1}\) and therefore \(\displaystyle{b}={\frac{{{1}}}{{{2}}}}\)
\(\displaystyle-{a}+{b}={0}\) gives us that \(\displaystyle{a}={b}\) and therefore \(\displaystyle{a}={\frac{{{1}}}{{{2}}}}\)
Hence \(\displaystyle{\left({w}\right)}_{{S}}={\left({\frac{{{1}}}{{{2}}}},{\frac{{{1}}}{{{2}}}}\right)}\)

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