The Given Set Is a Basis for a Subspace W

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.
-Let $\mathrm { x } _ { 1 } = \left[ \begin{array} { r } 0 \\ 1 \\ - 1 \\ 1 \end{array} \right] , \mathrm { x } _ { 2 } = \left[ \begin{array} { r } 1 \\ 1 \\ - 1 \\ - 1 \end{array} \right] , \mathrm { x } _ { 3 } = \left[ \begin{array} { l } 1 \\ 0 \\ 1 \\ 1 \end{array} \right]$

• A) \(\left[ \begin{array} { r } 0 \\ 1 \\ - 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \\ 4 \\ - 4 \\ - 2 \end{array} \right] , \left[ \begin{array} { r } 18 \\ 4 \\ 19 \\ 13 \end{array} \right]\)
• B)
• \(\left[ \begin{array} { r } 0 \\ 1 \\ - 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 1 \\ 0 \\ 0 \\ - 2 \end{array} \right] , \left[ \begin{array} { l } 6 \\ 0 \\ 1 \\ 3 \end{array} \right]\)
• C)
• \(\left[ \begin{array} { r } 0 \\ 1 \\ - 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 1 \\ 1 \\ - 1 \\ - 1 \end{array} \right] , \left[ \begin{array} { l } 1 \\ 0 \\ 1 \\ 1 \end{array} \right]\)
• D)
• \(\left[ \begin{array} { r } 0 \\ 1 \\ - 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \\ 2 \\ - 2 \\ - 4 \end{array} \right] , \left[ \begin{array} { r } 14 \\ 2 \\ 9 \\ 7 \end{array} \right]\)

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Definitions:

Female Preference Evolution

The evolutionary process by which female mating choices influence the development of male traits over generations.

Direct Benefits

Advantages that directly affect an individual's fitness or survival chances, often used in the context of mate selection where certain traits provide immediate advantages to the chooser.

Indirect Benefits

Advantages that do not directly influence an organism's survival or reproduction but may enhance the survival or reproductive success of its relatives or offspring.

Sexual Selection

A type of natural selection that arises through preference by one sex for certain characteristics in individuals of the other sex.