We are given the following ordered basis S for the vector space \(\displaystyle{V}={M}_{{22}}\) as well as the following vector v in V: \(S = \left\{ \begin{bmatrix}1 \\ 0 \end{bmatrix}\begin{bmatrix} 0 \\1 \end{bmatrix} \right\}; v = \begin{bmatrix} 3 \\-2 \end{bmatrix}\)

We wish to compute the coordinate vector \(\displaystyle{\left[{v}\right]}_{{s}}\) of v with respect to the basis S We have \([v]_s = \begin{bmatrix}a \\ b \end{bmatrix}\)

Where

\(v = \begin{bmatrix}3 \\ -2 \end{bmatrix}= a \begin{bmatrix}1 \\ 0 \end{bmatrix} + b \begin{bmatrix}0 \\ 1 \end{bmatrix}= \begin{bmatrix}a \\ b \end{bmatrix}\) Equating coefficients yields the following solution: \(\displaystyle{a}={1},\)

\(\displaystyle{b}=-{1},\)

\(\displaystyle{c}={0}:\)

\(\displaystyle{d}={2}.\) Therefore, the coordinate vector \(\displaystyle{\left[{v}\right]}_{{S}}\) of v with respect to the basis S is \([v]_S = \begin{bmatrix}3 \\ -2 \end{bmatrix}\)