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
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asked 2021-02-14

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 \(R^2, S = \left\{ \begin{bmatrix}1 \\ 0 \end{bmatrix}\begin{bmatrix} 0 \\1 \end{bmatrix} \right\}, v = \begin{bmatrix} 3 \\-2 \end{bmatrix} \)

Answers (1)

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

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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 \(R^2, S = \left\{ \begin{bmatrix}1 \\ 0 \end{bmatrix}\begin{bmatrix} 0 \\1 \end{bmatrix} \right\}, v = \begin{bmatrix} 3 \\-2 \end{bmatrix}\)

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