The vector x is in H = Span {v_{1}, v_{2}} and find the beta-coordinate vector [x]_{beta}

Question

The vector x is in

H = S p a n v_{1}, v_{2}

and find the beta-coordinate vector

[x]_{β}

Answer 1

Let vector x is in a vector space V and $β = b_{1}, b_{2}, \dots, b_{n}$ is a basis for V,
then the beta- coordinates of x are the weights $c_{1}, c_{2}, . . ., c_{n} o r [x]_{β} = β [\begin{matrix} c_{1} \\ c_{2} \\ \dots \\ c_{n} \end{matrix}] s u c h t h a t x = c_{1} b_{1} + c_{2} b_{2} + \dots + c_{n} b_{n} .$
Given:
The vectors $v_{1} = [\begin{matrix} 11 \\ - 5 \\ 10 \\ 7 \end{matrix}], v_{2} [\begin{matrix} 14 \\ - 8 \\ 13 \\ 10 \end{matrix}], a n d x = [\begin{matrix} 19 \\ - 13 \\ 18 \\ 15 \end{matrix}]$
Calculation:
Here $β = v_{1}, v_{2} a n d H = S p a n v_{1}, v_{2} .$
Therefore, for showing x is in H, show that the vector equation $x = x_{1} v_{1} + x_{2} v_{2}$ has a solution.
Write the given vectors as an augmented matrix $[v_{1} v_{2} x]$ and find the row reduced form of the matrix.
$[\begin{matrix} 11 & 14 & 19 \\ - 5 & - 8 & - 13 \\ 10 & 13 & 18 \\ 7 & 10 & 15 \end{matrix}] (R_{1} \to R_{1} + R_{2}) \vec{R_{3} \to R_{3} + 2 R_{2}} [\begin{matrix} 6 & 6 & 6 \\ - 5 & - 8 & - 13 \\ 0 & 3 & 8 \\ 7 & 10 & 15 \end{matrix}]$
$[\begin{matrix} 6 & 6 & 6 \\ - 5 & - 8 & - 13 \\ 0 & 3 & 8 \\ 7 & 10 & 15 \end{matrix}] R_{1} \to \frac{1}{6} R_{1} [\begin{matrix} 1 & 1 & 1 \\ - 5 & - 8 & - 13 \\ 0 & 3 & 8 \\ 7 & 10 & 15 \end{matrix}] [\begin{matrix} 1 & 1 & 1 \\ - 5 & - 8 & - 13 \\ 0 & 3 & 8 \\ 7 & 10 & 15 \end{matrix}] (R_{2} \to R_{2} - 5 R_{1}) \vec{R_{4} \to R_{4} - 7 R_{1}} [\begin{matrix} 1 & 1 & 1 \\ 0 & - 3 & - 8 \\ 0 & 3 & 8 \\ 0 & 3 & 8 \end{matrix}] [\begin{matrix} 1 & 1 & 1 \\ 0 & - 3 & - 8 \\ 0 & 3 & 8 \\ 0 & 3 & 8 \end{matrix}] (R_{3} \to R_{3} + R_{2}) \vec{R_{4} \to R_{4} + R_{2}} [\begin{matrix} 1 & 1 & 1 \\ 0 & - 3 & - 8 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$
On further simplification the matrix becomes,

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The vector x is in H = Span {v_{1}, v_{2}} and find the beta-coordinate vector [x]_{beta}

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