usagirl007A

2021-01-25

The vector x is in
and find the beta-coordinate vector $\left[x{\right]}_{\beta }$

likvau

Let vector x is in a vector space V and $\beta ={b}_{1},{b}_{2},\cdots ,{b}_{n}$ is a basis for V,
then the beta- coordinates of x are the weights
Given:
The vectors
Calculation:
Here
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 $\left[{v}_{1}{v}_{2}x\right]$ and find the row reduced form of the matrix.
$\left[\begin{array}{ccc}11& 14& 19\\ -5& -8& -13\\ 10& 13& 18\\ 7& 10& 15\end{array}\right]\left({R}_{1}\to {R}_{1}+{R}_{2}\right)\stackrel{\to }{{R}_{3}\to {R}_{3}+2{R}_{2}}\left[\begin{array}{ccc}6& 6& 6\\ -5& -8& -13\\ 0& 3& 8\\ 7& 10& 15\end{array}\right]$
$\left[\begin{array}{ccc}6& 6& 6\\ -5& -8& -13\\ 0& 3& 8\\ 7& 10& 15\end{array}\right]{R}_{1}\to \frac{1}{6}{R}_{1}\left[\begin{array}{ccc}1& 1& 1\\ -5& -8& -13\\ 0& 3& 8\\ 7& 10& 15\end{array}\right]\left[\begin{array}{ccc}1& 1& 1\\ -5& -8& -13\\ 0& 3& 8\\ 7& 10& 15\end{array}\right]\left({R}_{2}\to {R}_{2}-5{R}_{1}\right)\stackrel{\to }{{R}_{4}\to {R}_{4}-7{R}_{1}}\left[\begin{array}{ccc}1& 1& 1\\ 0& -3& -8\\ 0& 3& 8\\ 0& 3& 8\end{array}\right]\left[\begin{array}{ccc}1& 1& 1\\ 0& -3& -8\\ 0& 3& 8\\ 0& 3& 8\end{array}\right]\left({R}_{3}\to {R}_{3}+{R}_{2}\right)\stackrel{\to }{{R}_{4}\to {R}_{4}+{R}_{2}}\left[\begin{array}{ccc}1& 1& 1\\ 0& -3& -8\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
On further simplification the matrix becomes,

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