Question

# All bases considered in these are assumed to be ordered bases. In Exercise, compute thecoordinate vector of v with respect to the giving basis S for V.V is M_22

Alternate coordinate systems

All bases considered in these are assumed to be ordered bases. In Exercise, compute the coordinate vector of v with respect to the giving basis S for V. V is $$R^2, S = \left\{ \begin{bmatrix}1 \\ 0 \end{bmatrix}\begin{bmatrix} 0 \\1 \end{bmatrix} \right\}, v = \begin{bmatrix} 3 \\-2 \end{bmatrix}$$

2021-02-15

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 $$[v]_S = \begin{bmatrix}3 \\ -2 \end{bmatrix}$$