Q27E Determine whether the following ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q27E (page 143)

Determine whether the following vectors form a basis of ${鈩�}^{4}$ ; $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ - 1 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 4 \\ 8 \end{matrix}], [\begin{matrix} 1 \\ - 2 \\ 4 \\ - 8 \end{matrix}]$ .

Short Answer

Expert verified

The given vectors $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ - 1 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 4 \\ 8 \end{matrix}], [\begin{matrix} 1 \\ - 2 \\ 4 \\ - 8 \end{matrix}]$ form a basis of ${鈩�}^{4}$ .

Step by step solution

Definition of the basis of ℝn

The vectors $\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}, . . . . ., \overset{鈫�}{v_{n}}$ in ${鈩�}^{n}$ form a basis of ${鈩�}^{n}$ if (and only if) the matrix

$A = [\overset{鈫�}{v_{1}} \overset{鈫�}{v_{2}} . . . \overset{鈫�}{v_{n}}]$ is invertible.

Finding the determinant

Here we have $A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & - 1 & 2 & - 2 \\ 1 & 1 & 4 & 4 \\ 1 & - 1 & 8 & - 8 \end{matrix}]$

$鈬� |A| = 1 [\begin{matrix} - 1 & 2 & - 2 \\ 1 & 4 & 8 \\ - 1 & 8 & - 8 \end{matrix}] - 1 [\begin{matrix} 1 & 2 & - 2 \\ 1 & 4 & 4 \\ 1 & 8 & 8 \end{matrix}] + 1 [\begin{matrix} 1 & - 1 & - 2 \\ 1 & 1 & 4 \\ 1 & - 1 & - 8 \end{matrix}] - 1 [\begin{matrix} 1 & - 1 & 2 \\ 1 & 1 & 4 \\ 1 & - 1 & 8 \end{matrix}] 鈬� |A| = 72 - 0 - 12 - 4 = 56 鈬� |A| = 56 (鈮� 0)$

Since the matrix A is invertible, then the vectors $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ - 1 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 4 \\ 8 \end{matrix}], [\begin{matrix} 1 \\ - 2 \\ 4 \\ - 8 \end{matrix}]$ form a basis of ${鈩�}^{4}$ .

Final Answer

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91影视

Determine whether the following vectors form a basis of ${鈩�}^{4}$ ; $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ - 1 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 4 \\ 8 \end{matrix}], [\begin{matrix} 1 \\ - 2 \\ 4 \\ - 8 \end{matrix}]$ .

Short Answer

Step by step solution

Definition of the basis of ℝn

Finding the determinant

Final Answer

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91影视

Determine whether the following vectors form a basis of 鈩�4; [1111],[1-11-1],[1248],[1-24-8].

Short Answer

Step by step solution

Definition of the basis of ℝn

Finding the determinant

Final Answer

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Logic and Functions

Mechanics Maths

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Determine whether the following vectors form a basis of ${鈩�}^{4}$ ; $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ - 1 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 4 \\ 8 \end{matrix}], [\begin{matrix} 1 \\ - 2 \\ 4 \\ - 8 \end{matrix}]$ .