Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q3.2-54E (page 133)

Consider the line $L$ spanned by $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ in ${鈩�}^{3}$ . Find a basis of $L^{鈯�}$ . See Exercise 53.

Short Answer

Expert verified

The basis for $L^{鈯�}$ is, $\{[\begin{matrix} 3 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ - 1 \\ 0 \end{matrix}]\}$ .

Step by step solution

Consider the set

The vectors in ${鈩�}^{n}$ are linearly dependent if and only if there are nontrivial relation among them.

If linearly dependent, then, the vectors are said to be redundant.

Consider the vector $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$

Consider the condition

Let $\overset{鈫�}{w} 路 \overset{鈫�}{v} = 0, \overset{鈫�}{w} 鈭� L^{鈯�}, \overset{鈫�}{v} 鈭� V$

Then, $L^{鈯�}$ must be orthogonal to $L$ .

If, localid="1664203964228" $L = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ then, localid="1664203969891" $L^{鈯�} = [1 鈥� 鈥� 鈥� 鈥� 2 鈥� 鈥� 鈥� 鈥� 3]$

localid="1664203957436" $[1 鈥� 鈥� 鈥� 鈥� 2 鈥� 鈥� 鈥� 鈥� 3] 路 [x_{1} x_{2} x_{3}] = 0 x_{1} + 2 x_{2} + 3 x_{3} = 0$

Compute the values

If, $x_{3} = 0$

Then, $x_{1} = - 2 x_{2}$

Thus, $x_{1} = 2, x_{2} = - 1, x_{3} = 0$

$x = [2 - 1 0]$

If, $x_{2} = 0$

Then, $x_{1} = - 3 x_{3}$

Thus, $x_{1} = 3, x_{2} = 0, x_{3} = - 1$ .

$x = [3 0 - 1]$

Final answer

The basis for $L^{鈯�}$ is $\{[\begin{matrix} 3 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ - 1 \\ 0 \end{matrix}]\}$

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91影视

Consider the line $L$ spanned by $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ in ${鈩�}^{3}$ . Find a basis of $L^{鈯�}$ . See Exercise 53.

Short Answer

Step by step solution

Consider the set

Consider the condition

Compute the values

Final answer

One App. One Place for Learning.

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91影视

Consider the line Lspanned by[123]in鈩�3. Find a basis of L鈯�. See Exercise 53.

Short Answer

Step by step solution

Consider the set

Consider the condition

Compute the values

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Calculus

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Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Consider the line $L$ spanned by $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ in ${鈩�}^{3}$ . Find a basis of $L^{鈯�}$ . See Exercise 53.