Q20E In Exercises 10聽through 20聽, u... [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: Q20E (page 131)

In Exercises 10through 20, use paper and pencil to identify the redundant vectors. Thus determine whether the given vectors are linearly independent.
20 $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}], [\begin{matrix} 1 \\ 4 \\ 7 \\ 10 \end{matrix}] .$

Short Answer

Expert verified

The vectors $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}], [\begin{matrix} 1 \\ 4 \\ 7 \\ 10 \end{matrix}] .$ are redundant and linearly dependent.

Step by step solution

Consider the set.

The vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . . . . . . . {\overset{鈫�}{v}}_{m}$ 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 vectors, $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}], [\begin{matrix} 1 \\ 4 \\ 7 \\ 10 \end{matrix}] .$

Check for linear independency of vectors.

The reduced form of the matrix is,

$(\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 7 \\ 1 & 4 & 10 \end{matrix}) = (\begin{matrix} 1 & 1 & 1 \\ 0 & 3 & 9 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) = (\begin{matrix} 1 & 0 & - 2 \\ 00 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})$

Solve the above equations.

$v_{1} = [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], v_{2} = [\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}], v_{3} = [\begin{matrix} - 2 \\ 3 \\ 0 \\ 0 \end{matrix}] v_{3} = - 2 v_{1} + 3 v_{2}$

Final answer.

The vectors $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}], [\begin{matrix} 1 \\ 4 \\ 7 \\ 10 \end{matrix}] .$ are redundant and linearly dependent.

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91影视

In Exercises 10through 20, use paper and pencil to identify the redundant vectors. Thus determine whether the given vectors are linearly independent.
20 $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}], [\begin{matrix} 1 \\ 4 \\ 7 \\ 10 \end{matrix}] .$

Short Answer

Step by step solution

Consider the set.

Check for linear independency of vectors.

Final answer.

One App. One Place for Learning.

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91影视

In Exercises 10through 20, use paper and pencil to identify the redundant vectors. Thus determine whether the given vectors are linearly independent.20[1111],[1234],[14710].

Short Answer

Step by step solution

Consider the set.

Check for linear independency of vectors.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Statistics

Pure Maths

Decision Maths

Calculus

Study anywhere. Anytime. Across all devices.