The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.
-Let $x _ { 1 } = \left[ \begin{array} { r } 6 \ - 3 \ 0 \end{array} \right] , x _ { 2 } = \left[ \begin{array} { r } 6 \ - 18 \ 3 \end{array} \right]$
A)
$\left[ \begin{array} { r } 6 \ - 3 \ 0 \end{array} \right] , \left[ \begin{array} { r } 18 \ - 24 \ 3 \end{array} \right]$

B)
$\left[ \begin{array} { r } 6 \ - 3 \ 0 \end{array} \right] , \left[ \begin{array} { r } - 6 \ - 18 \ - 3 \end{array} \right]$

C)
$\left[ \begin{array} { r } - 9 \ - 3 \ 0 \end{array} \right] , \left[ \begin{array} { r } 6 \ - 18 \ 3 \end{array} \right]$

D)

$\left[ \begin{array} { r } 6 \ - 3 \ 0 \end{array} \right] , \left[ \begin{array} { r } - 6 \ - 12 \ 3 \end{array} \right]$

Question

The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.
-Let $x _ { 1 } = \left[ \begin{array} { r } 6 \ - 3 \ 0 \end{array} \right] , x _ { 2 } = \left[ \begin{array} { r } 6 \ - 18 \ 3 \end{array} \right]$
A)
$\left[ \begin{array} { r } 6 \ - 3 \ 0 \end{array} \right] , \left[ \begin{array} { r } 18 \ - 24 \ 3 \end{array} \right]$

B)
$\left[ \begin{array} { r } 6 \ - 3 \ 0 \end{array} \right] , \left[ \begin{array} { r } - 6 \ - 18 \ - 3 \end{array} \right]$

C)
$\left[ \begin{array} { r } - 9 \ - 3 \ 0 \end{array} \right] , \left[ \begin{array} { r } 6 \ - 18 \ 3 \end{array} \right]$

D)

$\left[ \begin{array} { r } 6 \ - 3 \ 0 \end{array} \right] , \left[ \begin{array} { r } - 6 \ - 12 \ 3 \end{array} \right]$

Quizplus · Accepted Answer

The Gram-Schmidt process starts with $x_1$ as the first vector of the orthogonal basis. The second vector is obtained by subtracting from $x_2$ its projection onto $x_1$. Calculation shows that the correct orthogonal vector to $x_1$ in the basis is $\left[ \begin{array} { r } - 6 \ - 12 \ 3 \end{array} \right]$, making option D correct.

The Given Set Is a Basis for a Subspace W x1=[6−30],x2=[6−183]x _ { 1 } = \left[ \begin{array} { r } 6 \\ - 3 \\ 0 \end{array} \right] , x _ { 2 } = \left[ \begin{array} { r } 6 \\ - 18 \\ 3 \end{array} \right]x1​=​6−30​​,x2​=​6−183​​

The Given Set Is a Basis for a Subspace W $x _ { 1 } = \left[ \begin{array} { r } 6 \\ - 3 \\ 0 \end{array} \right] , x _ { 2 } = \left[ \begin{array} { r } 6 \\ - 18 \\ 3 \end{array} \right]$