Question 19
Multiple Choice The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.x 1 = [ 0 1 − 1 1 ] , x 2 = [ 1 1 − 1 − 1 ] , x 3 = [ 1 0 1 1 ] \mathrm { x } _ { 1 } = \left[ \begin{array} { r } 0 \\ 1 \\ - 1 \\ 1 \end{array} \right] , \mathrm { x } _ { 2 } = \left[ \begin{array} { r } 1 \\ 1 \\ - 1 \\ - 1 \end{array} \right] , \mathrm { x } _ { 3 } = \left[ \begin{array} { l } 1 \\ 0 \\ 1 \\ 1 \end{array} \right] x 1 = 0 1 − 1 1 , x 2 = 1 1 − 1 − 1 , x 3 = 1 0 1 1
[ 0 1 − 1 1 ] , [ 3 4 − 4 − 2 ] , [ 18 4 19 13 ] \left[ \begin{array} { r } 0 \\ 1 \\ - 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \\ 4 \\ - 4 \\ - 2 \end{array} \right] , \left[ \begin{array} { r } 18 \\ 4 \\ 19 \\ 13 \end{array} \right] 0 1 − 1 1 , 3 4 − 4 − 2 , 18 4 19 13 [ 0 1 − 1 1 ] , [ 1 0 0 − 2 ] , [ 6 0 1 3 ] \left[ \begin{array} { r } 0 \\ 1 \\ - 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 1 \\ 0 \\ 0 \\ - 2 \end{array} \right] , \left[ \begin{array} { l } 6 \\ 0 \\ 1 \\ 3 \end{array} \right] 0 1 − 1 1 , 1 0 0 − 2 , 6 0 1 3 [ 0 1 − 1 1 ] , [ 1 1 − 1 − 1 ] , [ 1 0 1 1 ] \left[ \begin{array} { r } 0 \\ 1 \\ - 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 1 \\ 1 \\ - 1 \\ - 1 \end{array} \right] , \left[ \begin{array} { l } 1 \\ 0 \\ 1 \\ 1 \end{array} \right] 0 1 − 1 1 , 1 1 − 1 − 1 , 1 0 1 1 [ 0 1 − 1 1 ] , [ 3 2 − 2 − 4 ] , [ 14 2 9 7 ] \left[ \begin{array} { r } 0 \\ 1 \\ - 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \\ 2 \\ - 2 \\ - 4 \end{array} \right] , \left[ \begin{array} { r } 14 \\ 2 \\ 9 \\ 7 \end{array} \right] 0 1 − 1 1 , 3 2 − 2 − 4 , 14 2 9 7
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