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Linear Algebra With Applications
Found in: Page 132
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

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

Yes, the vectors v1,v1+v2,v1+v2+v3 are linearly independent.

See the step by step solution

Step by Step Solution

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

v1,v2,v3 are linearly independent vectors.

Since v1,v2,v3 we know that


From here


The equation is true only in case those are 0.

role="math" localid="1659358612562" x1v1+x2v1+v2+x3v1+v2+v3=0

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

Step 3: The final answer

Yes, the vectors v1,v1+v2,v1+v2+v3 are linearly independent.

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