Chapter 5: Q17E (page 216)

Find a basis for $W^{⊥}$ , where $W = s p a n {[\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}], [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}]}$ .

Short Answer

Expert verified

The orthogonal basis is $\vec{x} = s p a n \{[\begin{matrix} 1 \\ - 2 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ - 3 \\ 0 \\ 1 \end{matrix}]\}$ for $W = s p a n \{[\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}], [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}]\}$ .

Step by step solution

Determine the equation

Consider a set $W = s p a n \{[\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}], [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}]\}$ and $\vec{x} = [\begin{matrix} {\vec{x}}_{1} \\ {\vec{x}}_{2} \\ {\vec{x}}_{3} \\ {\vec{x}}_{4} \end{matrix}]$ is perpendicular to W .

If the vector $\vec{x}$ is perpendicular $\vec{y}$ then $\vec{x} . \vec{y} = 0$ .

As is perpendicular $\vec{x}$ to , by the definition of orthogonal $W . \vec{x} = 0$ .

Simplify the equation $\vec{x} . [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}] = 0$ as follows.

$\vec{x} . W = 0 [\begin{matrix} {\vec{x}}_{1} & {\vec{x}}_{2} & {\vec{x}}_{3} & {\vec{x}}_{4} \end{matrix}] [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}] = 0 \begin{matrix} {\vec{x}}_{1} + & 2 {\vec{x}}_{2} + & 3 {\vec{x}}_{3} + & 4 {\vec{x}}_{4} \end{matrix} = 0$

Simplify the equation $\vec{x} . [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}] = 0$ as follows.

$[\begin{matrix} {\vec{x}}_{1} & {\vec{x}}_{2} & {\vec{x}}_{3} & {\vec{x}}_{4} \end{matrix}] . W = 0 [\begin{matrix} {\vec{x}}_{1} & {\vec{x}}_{2} & {\vec{x}}_{3} & {\vec{x}}_{4} \end{matrix}] [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}] = 0 \begin{matrix} 5 {\vec{x}}_{1} + & 6 {\vec{x}}_{2} + & 7 {\vec{x}}_{3} + & 8 {\vec{x}}_{4} \end{matrix} = 0$

Determine the orthogonal basis

Simplify the equations $5 {\vec{x}}_{1} + 6 {\vec{x}}_{2} + 7 {\vec{x}}_{3} + 8 {\vec{x}}_{4} = 0$ and ${\vec{x}}_{1} + 2 {\vec{x}}_{2} + 3 {\vec{x}}_{3} + 4 {\vec{x}}_{4} = 0$ as follows.

role="math" localid="1659928785801" ${\vec{x}}_{1} + 2 {\vec{x}}_{2} + 3 {\vec{x}}_{3} + 4 {\vec{x}}_{4} = 0 - \{2 {\vec{x}}_{2} + 3 {\vec{x}}_{3} + 4 {\vec{x}}_{4}\} = {\vec{x}}_{1}$

Substitute the value $\{2 {\vec{x}}_{2} + 3 {\vec{x}}_{3} + 4 {\vec{x}}_{4}\}$ for ${\vec{x}}_{1}$ in the equation $5 {\vec{x}}_{1} + 6 {\vec{x}}_{2} + 7 {\vec{x}}_{3} + 8 {\vec{x}}_{4} = 0$ as follows.

$5 {\vec{x}}_{1} + 6 {\vec{x}}_{2} + 7 {\vec{x}}_{3} + 8 {\vec{x}}_{4} = 0 5 . - \{2 {\vec{x}}_{2} + 3 {\vec{x}}_{3} + 4 {\vec{x}}_{4}\} + 6 {\vec{x}}_{2} + 7 {\vec{x}}_{3} + 8 {\vec{x}}_{4} = 0 - \{10 {\vec{x}}_{2} + 15 {\vec{x}}_{3} + 20 {\vec{x}}_{4}\} + 6 {\vec{x}}_{2} + 7 {\vec{x}}_{3} + 8 {\vec{x}}_{4} = 0 - 4 {\vec{x}}_{2} - 8 {\vec{x}}_{3} - 12 {\vec{x}}_{4} = 0$

Simplify the equation as follows:

$- 4 {\vec{x}}_{2} - 8 {\vec{x}}_{3} - 12 {\vec{x}}_{4} = 0 - {\vec{x}}_{2} - {\vec{x}}_{3} - {\vec{x}}_{4} = 0 - 2 {\vec{x}}_{3} - 3 {\vec{x}}_{4} = {\vec{x}}_{2}$

Substitute the valuerole="math" localid="1659929137563" $- 2 {\vec{x}}_{3} - 3 {\vec{x}}_{4}$ for ${\vec{x}}_{2}$ in the equation ${\vec{x}}_{1} + 2 {\vec{x}}_{2} + 3 {\vec{x}}_{3} + 4 {\vec{x}}_{4} = 0$ as follows:

${\vec{x}}_{1} + 2 {\vec{x}}_{2} + 3 {\vec{x}}_{3} + 4 {\vec{x}}_{4} = 0 {\vec{x}}_{1} + 2 \{- 2 {\vec{x}}_{3} + 3 {\vec{x}}_{4}\} + 3 {\vec{x}}_{3} + 4 {\vec{x}}_{4} = 0 {\vec{x}}_{1} + \{- 4 {\vec{x}}_{3} + 6 {\vec{x}}_{4}\} + 3 {\vec{x}}_{3} + 4 {\vec{x}}_{4} = 0 {\vec{x}}_{1} + {\vec{x}}_{3} + 2 {\vec{x}}_{4} = 0$

Further, simplify the equation as follows:

${\vec{x}}_{1} + {\vec{x}}_{3} + 2 {\vec{x}}_{4} = 0 {\vec{x}}_{1} = {\vec{x}}_{3} + 2 {\vec{x}}_{4}$

Substitute the values $- 2 {\vec{x}}_{3} - 3 {\vec{x}}_{4}$ for ${\vec{x}}_{2}$ and ${\vec{x}}_{3} + 2 {\vec{x}}_{4}$ for ${\vec{x}}_{1}$ in the equation $\vec{x} = [\begin{matrix} {\vec{x}}_{1} \\ {\vec{x}}_{2} \\ {\vec{x}}_{3} \\ {\vec{x}}_{4} \end{matrix}]$ as follows:

$\vec{x} = [\begin{matrix} {\vec{x}}_{1} \\ {\vec{x}}_{2} \\ {\vec{x}}_{3} \\ {\vec{x}}_{4} \end{matrix}] \vec{x} = [\begin{matrix} {\vec{x}}_{3} + 2 {\vec{x}}_{4} \\ - 2 {\vec{x}}_{3} - 3 {\vec{x}}_{4} \\ {\vec{x}}_{3} \\ {\vec{x}}_{4} \end{matrix}] \vec{x} = [\begin{matrix} {\vec{x}}_{3} \\ - 2 {\vec{x}}_{3} \\ {\vec{x}}_{3} \\ 0 \end{matrix}] + [\begin{matrix} 2 {\vec{x}}_{4} \\ - 3 {\vec{x}}_{4} \\ 0 \\ {\vec{x}}_{4} \end{matrix}] \vec{x} = {\vec{x}}_{3} [\begin{matrix} 1 \\ - 2 \\ 1 \\ 0 \end{matrix}] + {\vec{x}}_{4} [\begin{matrix} 2 \\ - 3 \\ 0 \\ 1 \end{matrix}]$

Therefore, the orthogonal basis is $\vec{x} = s p a n \{[\begin{matrix} 1 \\ - 2 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ - 3 \\ 0 \\ 1 \end{matrix}]\}$ .

Hence, the orthogonal basis is $\vec{x} = s p a n \{[\begin{matrix} 1 \\ - 2 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ - 3 \\ 0 \\ 1 \end{matrix}]\}$ for $W = s p a n \{[\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}], [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}]\}$ .

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Find a basis for $W^{⊥}$ , where $W = s p a n {[\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}], [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}]}$ .

Short Answer

Step by step solution

Determine the equation

Determine the orthogonal basis

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Find a basis forW⊥, whereW=span{1234,5678}.

Short Answer

Step by step solution

Determine the equation

Determine the orthogonal basis

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Statistics

Theoretical and Mathematical Physics

Probability and Statistics

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Find a basis for $W^{⊥}$ , where $W = s p a n {[\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}], [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}]}$ .