Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Consider some linearly independent vectors in and a vector ${\vec{v}}_{1,} {\vec{v}}_{2,} . . . {\vec{v}}_{m,}$ in $ℝ^{n} a n d a v e c t o r \vec{v} i n ℝ^{n}$ that is not contained in the span ${\vec{v}}_{1}, \vec{v_{2}}, . . . {\vec{v}}_{m}$ of . Are the vectors ${\vec{v}}_{1,} {\vec{v}}_{2,} . . . . {\vec{v}}_{m}, {\vec{v}}_{}$ necessarily linearly independent?

If there are some linearly independent vectors in and a vector ${\vec{v}}_{1}, {\vec{v}}_{2}, . . {\vec{v}}_{m},$ in $ℝ^{n} a n d a v e c t o r \vec{v} i n ℝ^{n}$ that is not contained in the span of ${\vec{v}}_{1}, {\vec{v}}_{2}, . . . {\vec{v}}_{m}$ then the vectors are necessarily linearly independent.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Linear Relations

Consider the vectors ${\vec{v}}_{1}, {\vec{v}}_{2}, . . . {\vec{v}}_{m}$ in . An equation of the form

$a_{1} {\vec{v}}_{1} + a_{2} {\vec{v}}_{2} + . . . + a_{m} {\vec{v}}_{m} = 0$ (1)

is called a (linear) relation among the vectors ${\vec{v}}_{1}, {\vec{v}}_{2}, . . . {\vec{v}}_{3}$ . There is always the trivial relation, with $a_{1} = 0 = a_{2} = . . . = a_{m}$ . Nontrivial relations (where at least one coefficient $a_{i}$ is nonzero) may or may not exist among the vectors ${\vec{v}}_{1}, {\vec{v}}_{2}, . . . {\vec{v}}_{m} .$ .

Step 2: To prove the vectors v→1,v→2,...v→m,v→ are linearly independent

Since the vector $\vec{v}$ in $ℝ^{n}$ is not contained in the span of ${\vec{v}}_{1}, {\vec{v}}_{2}, . . . {\vec{v}}_{m},$ . Therefore

$\vec{v} \neq a_{1} {\vec{v}}_{1}, + a_{2} {\vec{v}}_{2} + . . . + a_{m} {\vec{v}}_{m}$ , for $a_{i} \neq 0$ . (2)

Now multiply the equation (2) by role="math" localid="1659416449027" $b \in ℝ a n d b \neq 0$ from left ,then we have

$b \vec{v} \neq b a_{1} {\vec{v}}_{1} + b a_{2} {\vec{v}}_{2} + . . . + b a_{m} {\vec{v}}_{m}$ (3)

Now subtract both sides by $b \vec{v},$ , we get

role="math" localid="1659416709972" $0 \neq b a_{1} {\vec{v}}_{1} + b a_{2} {\vec{v}}_{2} + . . . b a_{m} {\vec{v}}_{m} - b \vec{v}$ (4)

Since $b \neq 0 \Rightarrow b a_{i} \neq 0$ , so the right hand side of equation (4) will be zero if and only if $b a_{i} = 0$ .

Hence, the vectors ${\vec{v}}_{1}, {\vec{v}}_{2}, . . . {\vec{v}}_{m} {\vec{v}}_{}$ are linearly independent.

Step 3: Final Answer

If there are some linearly independent vectors in ${\vec{v}}_{1}, {\vec{v}}_{2}, . . . {\vec{v}}_{m} i n ℝ^{n}$ and a vector $ℝ^{n}$ in that is not contained in the span ${\vec{v}}_{1}, {\vec{v}}_{2}, . . . {\vec{v}}_{m}$ of then the vectors ${\vec{v}}_{1}, {\vec{v}}_{2}, . . . {\vec{v}}_{m}, \vec{v}$ are necessarily linearly independent.

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

Step 1: Linear Relations

Step 2: To prove the vectors v→1,v→2,...v→m,v→ are linearly independent

Step 3: Final Answer

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Linear Algebra With Applications

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Short Answer

Consider some linearly independent vectors in and a vector v→1,v→2,...v→m, in ℝn and a vectorv→ in ℝn that is not contained in the span v→1,v2→,...v→m of . Are the vectors v→1,v→2,....v→m,v→ necessarily linearly independent?

Step by Step Solution

Step 1: Linear Relations

Step 2: To prove the vectors v→1,v→2,...v→m,v→ are linearly independent

Step 3: Final Answer

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