Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q32E (page 144)

Find the basis of subspace of ${鈩�}^{4}$ that consists of all vectors perpendicular to both
$[\begin{matrix} 1 \\ 0 \\ - 1 \\ 1 \end{matrix}]$ and $[\begin{matrix} 0 \\ 1 \\ 2 \\ 3 \end{matrix}]$ .
See definition A.8 in the Appendix.

Short Answer

Expert verified

The required basis is,

$[\begin{matrix} 1 \\ 鈭� 2 \\ 1 \\ 0 \end{matrix}], 鈥夆赌� [\begin{matrix} 1 \\ 3 \\ 0 \\ 鈭� 1 \end{matrix}]$ .

Step by step solution

To recall the definition A.8 from Appendix

From the Definition A.8 in Appendix, we have,

鈥淭wo vectors $\overset{鈫�}{蠀}$ and $\overset{鈫�}{蝇}$ in ${鈩�}^{n}$ are called perpendicular (or orthogonal) if $\overset{鈫�}{蠀} 鈰� \overset{鈫�}{蝇} = 0$ .鈥�

We need to find the vectors $\overset{鈫�}{x} 鈭� {鈩�}^{4}$ such that their scalar product is 0.

$[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] 鈰� [\begin{matrix} 1 \\ 0 \\ 鈭� 1 \\ 1 \end{matrix}] = 0$ and $[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] 鈰� [\begin{matrix} 0 \\ 1 \\ 2 \\ 3 \end{matrix}] = 0$ .

To find the basis of subspace of ℝ4

On solving the above product of matrices, we get,

$\begin{array}{l} [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] 鈰� [\begin{matrix} 1 \\ 0 \\ 鈭� 1 \\ 1 \end{matrix}] = 0 \\ 鈬� x_{1} 鈭� x_{3} + x_{4} = 0 \end{array}$

and

$\begin{array}{l} [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] 鈰� [\begin{matrix} 0 \\ 1 \\ 2 \\ 3 \end{matrix}] = 0 \\ 鈬� x_{2} + 2 x_{3} + 3 x_{4} = 0 \end{array}$

Therefore, we get the matrix $[\begin{matrix} 1 & 0 & 鈭� 1 & 1 \\ 0 & 1 & 2 & 3 \end{matrix}]$

It means to find the kernel of that matrix.

Thus, the redundant vectors are given by,

$\begin{matrix} 蠀_{3} = 鈭� 蠀_{1} + 2 蠀_{2} 鈬� 蠀_{1} 鈭� 2 蠀_{2} + 蠀_{3} = 0 \\ 蠀_{4} = 蠀_{1} + 3 蠀_{2} 鈬� 蠀_{1} + 3 蠀_{2} 鈭� 蠀_{4} = 0 \end{matrix}$

Hence, the basis is given by,

$[\begin{matrix} 1 \\ 鈭� 2 \\ 1 \\ 0 \end{matrix}], 鈥夆赌� [\begin{matrix} 1 \\ 3 \\ 0 \\ 鈭� 1 \end{matrix}]$ .

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91影视

Find the basis of subspace of ${鈩�}^{4}$ that consists of all vectors perpendicular to both
$[\begin{matrix} 1 \\ 0 \\ - 1 \\ 1 \end{matrix}]$ and $[\begin{matrix} 0 \\ 1 \\ 2 \\ 3 \end{matrix}]$ .
See definition A.8 in the Appendix.

Short Answer

Step by step solution

To recall the definition A.8 from Appendix

To find the basis of subspace of ℝ4

One App. One Place for Learning.

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91影视

Find the basis of subspace of 鈩�4that consists of all vectors perpendicular to both[10-11]and[0123] .See definition A.8 in the Appendix.

Short Answer

Step by step solution

To recall the definition A.8 from Appendix

To find the basis of subspace of ℝ4

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Mechanics Maths

Discrete Mathematics

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find the basis of subspace of ${鈩�}^{4}$ that consists of all vectors perpendicular to both
$[\begin{matrix} 1 \\ 0 \\ - 1 \\ 1 \end{matrix}]$ and $[\begin{matrix} 0 \\ 1 \\ 2 \\ 3 \end{matrix}]$ .
See definition A.8 in the Appendix.