Q43E Question: Consider three linearl... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q43E (page 132)

Question: Consider three linearly independent vectors $\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}, \overset{鈫�}{v_{3}}$ in ${鈩�}^{3}$ . Are the vectors $\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{1}} + \overset{鈫�}{v_{2}}, \overset{鈫�}{v_{1}} + \overset{鈫�}{v_{2}} + \overset{鈫�}{v_{3}}$ linearly independent as well? How can you tell?

Short Answer

Expert verified

Yes, the vectors $\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{1}} + \overset{鈫�}{v_{2}}, \overset{鈫�}{v_{1}} + \overset{鈫�}{v_{2}} + \overset{鈫�}{v_{3}}$ are linearly independent.

Step by step solution

Definition of Linearly independent

Linearly independent is defined as the property of a set having no linear combination of its elements equal to zero when the coefficients are taken from a given set unless the coefficient of each element is zero.

Prove the vectors are linearly independent

$\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}, \overset{鈫�}{v_{3}}$ are linearly independent vectors.

Since $\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}, \overset{鈫�}{v_{3}}$ we know that

$(x_{1}, x_{2}, x_{3}) \overset{鈫�}{v_{1}}, (x_{2}, x_{3}) \overset{鈫�}{v_{2}}, x_{3} \overset{鈫�}{v_{3}} = 0 鈬� (x_{1}, x_{2}, x_{3}) = (x_{2}, x_{3}) = x_{3} = 0$

From here

$x_{3} = 0 x_{1} + x_{2} = 0 鈫� x_{2} = 0 x_{1}, x_{2}, x_{3} = 0 鈫� x_{1} = 0$

The equation is true only in case those are 0.

role="math" localid="1659358612562" $x_{1} \overset{鈫�}{v_{1}} + x_{2} (\overset{鈫�}{v_{1}} + \overset{鈫�}{v_{2}}) + x_{3} (\overset{鈫�}{v_{1}} + \overset{鈫�}{v_{2}} + \overset{鈫�}{v_{3}}) = 0$

From here it is clear that the vectors are linearly independent.

The final answer

Yes, the vectors $\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{1}} + \overset{鈫�}{v_{2}}, \overset{鈫�}{v_{1}} + \overset{鈫�}{v_{2}} + \overset{鈫�}{v_{3}}$ are linearly independent.

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91影视

Short Answer

Step by step solution

Definition of Linearly independent

Prove the vectors are linearly independent

The final answer

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91影视

Question: Consider three linearly independent vectorsv1鈫�,v2鈫�,v3鈫�in 鈩�3. Are the vectorsv1鈫�,v1鈫�+v2鈫�,v1鈫�+v2鈫�+v3鈫�linearly independent as well? How can you tell?

Short Answer

Step by step solution

Definition of Linearly independent

Prove the vectors are linearly independent

The final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Logic and Functions

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.