Mathematics, 31.08.2019 04:30 Queenashley3232
Use a linear system to write u = (12, 19, 31) as a linear combination of u1 = (1,1,2), u2 = (2,3,5), and uz = (3,5,8). is w = (1,0,1) a linear com- bination of u1, u2, u3?
Answers: 1
,
.
is also a linear combination of 
, 
, 
.
, with one equation for each component. The augmented matrix for the first linear system will be:![\displaystyle \left[\begin{array}{ccc|c}1 & 2 & 3 & 12\\1 & 3 & 5 & 19\\2 & 5 & 8 & 31\end{array}\right]](/tpl/images/0213/9895/d4cfa.png)
.![\begin{aligned} &\left[\begin{array}{ccc|c}1 & 2 & 3 & 12\\1 & 3 & 5 & 19\\2 & 5 & 8 & 31\end{array}\right]\\ &\sim \left[\begin{array}{ccc|c}1 & 2 & 3 & 12\\0 & 1 & 2 & 7\\0 & 1 & 2 & 7\end{array}\right]\\&\sim \left[\begin{array}{ccc|c}1 & 2 & 3 & 12\\0 & 1 & 2 & 7\\0 & 0 & 0 & 0\end{array}\right]\\&\sim\left[\begin{array}{ccc|c}1 & 0 & -1 & -2\\0 & 1 & 2 & 7\\0 & 0 & 0 & 0\end{array}\right]\\&\left\{\begin{array}{l}x_1 = t-2\\x_2=- 2t+7\\x_3=t\end{array}\right.\end{aligned}](/tpl/images/0213/9895/70f61.png)
.
:![\left[\begin{array}{ccc|c}1 & 2 & 3 & 1\\1 & 3 & 5 & 0\\2 & 5 & 8 & 1\end{array}\right]](/tpl/images/0213/9895/053f0.png)
.
, 
, or 
. Therefore, before proceeding with Gaussian Elimination, start by checking the determinant of the coefficient matrix. If this determinant is nonzero, ![\begin{aligned} &\left[\begin{array}{ccc|c}1 & 2 & 3 & 1\\1 & 3 & 5 & 0\\2 & 5 & 8 & 1\end{array}\right]\\ &\sim \left[\begin{array}{ccc|c}1 & 2 & 3 & 1\\0 & 1 & 2 & -1\\0 & 1 & 2 & -1\end{array}\right]\\&\sim \left[\begin{array}{ccc|c}1 & 2 & 3 & 1\\0 & 1 & 2 & -1\\0 & 0 & 0 & 0\end{array}\right]\\&\sim\left[\begin{array}{ccc|c}1 & 0 & -1 & 3\\0 & 1 & 2 & -1\\0 & 0 & 0 & 0\end{array}\right]\\&\left\{\begin{array}{l}x_1 = t+3\\x_2=- 2t-1\\x_3=t\end{array}\right.\end{aligned}](/tpl/images/0213/9895/6b84a.png)
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Use a linear system to write u = (12, 19, 31) as a linear combination of u1 = (1,1,2), u2 = (2,3,5),...
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