# Find the coordinate vector of w relative to the basis

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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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 $$\displaystyle{\left({w}\right)}_{{S}}={\left({\frac{{{1}}}{{{2}}}},{\frac{{{1}}}{{{2}}}}\right)}$$