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Expert-verified Found in: Page 132 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Show that there is a nontrivial relation among the vectors ${\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{1}}}{\mathbf{,}}{\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{2}}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{,}}{\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{m}}}$ if (and only if) at least one of the vectors is a linear combination of the other vectors${\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{1}}}{\mathbf{,}}{\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{2}}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\stackrel{\mathbf{\to }}{\mathbf{v}}}_{\mathbf{i}\mathbf{-}\mathbf{1}}{\mathbf{,}}{\stackrel{\mathbf{\to }}{\mathbf{v}}}_{\mathbf{i}\mathbf{+}\mathbf{1}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{,}}{\stackrel{\mathbf{\to }}{\mathbf{v}}}_{{\mathbf{m}}}{\mathbf{.}}$

There is a nontrivial relation among the vectors ${\stackrel{\to }{\mathrm{v}}}_{1},{\stackrel{\to }{\mathrm{v}}}_{2},...,{\stackrel{\to }{\mathrm{v}}}_{\mathrm{m}}$ if (and only if) at least one of the vectors ${\stackrel{\to }{\mathrm{v}}}_{\mathrm{i}}$ is a linear combination of the other vectors ${\stackrel{\to }{\mathrm{v}}}_{1},{\stackrel{\to }{\mathrm{v}}}_{2},....{\stackrel{\to }{\mathrm{v}}}_{\mathrm{i}-1},{\stackrel{\to }{\mathrm{v}}}_{\mathrm{i}+1},....,{\stackrel{\to }{\mathrm{v}}}_{\mathrm{m}}.$ .

See the step by step solution

## Step 1: Writing the vectors v→i in a linear combination of the other vectors

Let us suppose that there is a non-trivial relation

${c}_{1}v{⃗}_{1}+{c}_{2}v{⃗}_{2}+...+{c}_{i-1}v{⃗}_{i-1}+{c}_{i}v{⃗}_{i}+{c}_{i+1}v{⃗}_{i+1}+...+{c}_{m}v{⃗}_{m}=0$ with ${c}_{i}\ne 0,i=1,2,...,m$

Then we have

${c}_{i}v{⃗}_{i}=-{c}_{1}v{⃗}_{1}-{c}_{2}v{⃗}_{2}-...-{c}_{i-1}v{⃗}_{i-1}-{c}_{i-1}v{⃗}_{i+1}-...-{c}_{m}v{⃗}_{m}$

$⇒{\stackrel{\to }{v}}_{i}=-{c}_{i}^{-1}{c}_{1}{\stackrel{\to }{v}}_{1}-{c}_{i}^{-1}{c}_{2}{\stackrel{\to }{v}}_{2}-.....-{c}_{i}^{-1}{c}_{i-1}{\stackrel{\to }{v}}_{i-1}-{c}_{i}^{-1}{c}_{i+1}{\stackrel{\to }{v}}_{i+1}-...-{c}_{i}^{-1}{c}_{m}{\stackrel{\to }{v}}_{m}$

The vector is a linear combination of the other vector .

## Step 2:   To prove the converse part of the given statement

Conversely, let us consider that the vector is a linear combination of the other vectors ${\stackrel{\to }{\mathrm{v}}}_{1},{\stackrel{\to }{\mathrm{v}}}_{2},....{\stackrel{\to }{\mathrm{v}}}_{\mathrm{i}-1},{\stackrel{\to }{\mathrm{v}}}_{\mathrm{i}+1},....,{\stackrel{\to }{\mathrm{v}}}_{\mathrm{m}}.$ . Then we can write

${\stackrel{\to }{\mathrm{v}}}_{1}={d}_{1}{\stackrel{\to }{\mathrm{v}}}_{1}+{d}_{2}{\stackrel{\to }{\mathrm{v}}}_{2}+...+{d}_{i-1}{\stackrel{\to }{\mathrm{v}}}_{i-1}+{d}_{i+1}{\stackrel{\to }{\mathrm{v}}}_{i+1}+...+{d}_{m}{\stackrel{\to }{\mathrm{v}}}_{m}$

Subtracting role="math" localid="1659359195199" ${\stackrel{\to }{\mathrm{v}}}_{i}$ from both sides of above equation, we get

role="math" localid="1659359178216" $0={d}_{1}{\stackrel{\to }{\mathrm{v}}}_{1}+{d}_{2}{\stackrel{\to }{\mathrm{v}}}_{2}+...+{d}_{i-1}{\stackrel{\to }{\mathrm{v}}}_{i-1}+{d}_{i+1}{\stackrel{\to }{\mathrm{v}}}_{i+1}+...+{d}_{m}{\stackrel{\to }{\mathrm{v}}}_{m}$

Thus, we get a non-trivial relation among the vectors ${\stackrel{\to }{\mathrm{v}}}_{1},{\stackrel{\to }{\mathrm{v}}}_{2},...,{\stackrel{\to }{\mathrm{v}}}_{\mathrm{m}}.$ if the vector is a linear combination of the other vectors ${\stackrel{\to }{\mathrm{v}}}_{1},{\stackrel{\to }{\mathrm{v}}}_{2},....{\stackrel{\to }{\mathrm{v}}}_{\mathrm{i}-1},{\stackrel{\to }{\mathrm{v}}}_{\mathrm{i}+1},....,{\stackrel{\to }{\mathrm{v}}}_{\mathrm{m}}.$ .

There is a nontrivial relation among the vectors ${\stackrel{\to }{\mathrm{v}}}_{1},{\stackrel{\to }{\mathrm{v}}}_{2},...,{\stackrel{\to }{\mathrm{v}}}_{\mathrm{m}}.$ if (and only if) at least one of the vectors ${\stackrel{\to }{\mathrm{v}}}_{i}$ is a linear combination of the other vectors ${\stackrel{\to }{\mathrm{v}}}_{1},{\stackrel{\to }{\mathrm{v}}}_{2},....{\stackrel{\to }{\mathrm{v}}}_{\mathrm{i}-1},{\stackrel{\to }{\mathrm{v}}}_{\mathrm{i}+1},....,{\stackrel{\to }{\mathrm{v}}}_{\mathrm{m}}.$ . ### Want to see more solutions like these? 