Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: 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.

Step 2: Prove the vectors are linearly independent

$\vec{v_{1}}, \vec{v_{2}}, \vec{v_{3}}$ are linearly independent vectors.

Since $\vec{v_{1}}, \vec{v_{2}}, \vec{v_{3}}$ we know that

$(x_{1}, x_{2}, x_{3}) \vec{v_{1}}, (x_{2}, x_{3}) \vec{v_{2}}, x_{3} \vec{v_{3}} = 0 \Rightarrow (x_{1}, x_{2}, x_{3}) = (x_{2}, x_{3}) = x_{3} = 0$

From here

$x_{3} = 0 x_{1} + x_{2} = 0 \to x_{2} = 0 x_{1}, x_{2}, x_{3} = 0 \to x_{1} = 0$

The equation is true only in case those are 0.

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

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

Step 3: The final answer

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

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Linear Algebra With Applications

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Short Answer

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

Step by Step Solution

Step 1: Definition of Linearly independent

Step 2: Prove the vectors are linearly independent

Step 3: The final answer

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Linear Algebra With Applications

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Short Answer

Question: Consider three linearly independent vectors v1→,v2→,v3→ in ℝ3. Are the vectors v1→,v1→+v2→,v1→+v2→+v3→ linearly independent as well? How can you tell?

Step by Step Solution

Step 1: Definition of Linearly independent

Step 2: Prove the vectors are linearly independent

Step 3: The final answer

Most popular questions for Math Textbooks

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