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Example $$1 :$$ Let $$S=\left\{v_{1}, v_{2}, v_{3}, v_{4}\right\}$$ be a basis for $$R^{4},$$ where:

$$\begin{array}{ll}{v_{1}=(1,1,0,0)} & {v_{2}=(2,0,1,0)} \\ {v_{3}=(0,1,2,-1)} & {v_{4}=(0,1,-1,0)}\end{array} $$

if $$v=(1,2,-6,2)$$

Compute $$[v]_{s}$$

$$V=c_{1} v_{1}+c_{2} v_{2}+c_{3} v_{3}+c_{4} v_{4} $$

$$\left(\begin{array}{cccc|c}{1} & {2} & {0} & {0} & {1} \\ {1} & {0} & {1} & {1} & {2} \\ {0} & {1} & {2} & {-1} & {-6} \\ {0} & {0} & {-1} & {0} & {2}\end{array}\right)$$

$$\Rightarrow c_{1}=3, c_{2}=-1, c_{3}=-2, c_{4}=1$$

$$[v]_{s}=\left[\begin{array}{c}{3} \\ {-1} \\ {-2} \\ {1}\end{array}\right]$$

Example $$2 :$$ Let $$S=\left\{e_{1}, e_{2}, e_{3}\right\}$$ be the natural basis
for $$R^{3},$$ and let
$$v=(2,-1,3)$$
Compute $$[v]_{s}$$

$$V=c_{1} e_{1}+c_{2} e_{2}+c_{3} e_{3} $$

$$V=2 e_{1}-e_{2}+3 e_{3} $$

$$[v]_{s}=\left[\begin{array}{c}{2} \\ {-1} \\ {3}\end{array}\right]$$

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