Q9E Consider the vectorsv鈫�1,v鈫�2,... [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 1: Q9E (page 1)

Consider the vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . . ., {\overset{鈫�}{v}}_{m} in {鈩�}^{n} with {\overset{鈫�}{v}}_{m} = \overset{鈫�}{0}$ . Are these vectors linearly independent?

Short Answer

Expert verified

The vectors $({\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . . ., {\overset{鈫�}{v}}_{m}) with {\overset{鈫�}{v}}_{m} = \overset{鈫�}{0}$ are linearly dependent.

Step by step solution

Consider the set.

The vectors $({\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . . ., {\overset{鈫�}{v}}_{m})$ are linearly independent if for any linear combination of these vectors say $C_{1} {\overset{鈫�}{v}}_{1}, C_{2} {\overset{鈫�}{v}}_{2}, . . . . ., C_{m} {\overset{鈫�}{v}}_{m} = \overset{鈫�}{0}$ , the coefficient for these vectors are zero that is $C_{1} = 0, C_{2} = 0, . . . . ., C_{m} = 0$ .

Consider the vectors $({\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . . ., {\overset{鈫�}{v}}_{m}) in {鈩�}^{n} wth {\overset{鈫�}{v}}_{m} = \overset{鈫�}{0}$ in with .

Check for condition.

Let $C_{1} {\overset{鈫�}{v}}_{1} + C_{2} {\overset{鈫�}{v}}_{2} + . . . . . + C_{m} {\overset{鈫�}{v}}_{m} = \overset{鈫�}{0} . . . . . . (1)$

For $c_{1} = 0, c_{2} = 0, . . . . . . c_{m - 1} = 0 and c_{m} = 2$ the equation ( 1 ) holds good.

This violates that these are linearly independent set.

Final answer.

The vectors $({\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . . ., {\overset{鈫�}{v}}_{m}) with {\overset{鈫�}{v}}_{m} = \overset{鈫�}{0}$ are linearly dependent.

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91影视

Consider the vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . . ., {\overset{鈫�}{v}}_{m} in {鈩�}^{n} with {\overset{鈫�}{v}}_{m} = \overset{鈫�}{0}$ . Are these vectors linearly independent?

Short Answer

Step by step solution

Consider the set.

Check for condition.

Final answer.

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91影视

Consider the vectorsv鈫�1,v鈫�2,.....,v鈫�min鈩�nwithv鈫�m=0鈫�. Are these vectors linearly independent?

Short Answer

Step by step solution

Consider the set.

Check for condition.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Consider the vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . . ., {\overset{鈫�}{v}}_{m} in {鈩�}^{n} with {\overset{鈫�}{v}}_{m} = \overset{鈫�}{0}$ . Are these vectors linearly independent?