Question

Show that there is a nontrivial relation among the vectors ${\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{m}}$if (and only if) at least one of the vectorsis a linear combination of the other vectors

${\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\overrightarrow{\mathbf{v}}}_{\mathbf{i}\mathbf{-}\mathbf{1}}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{i}\mathbf{+}\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{m}}\mathbf{.}$

Short answer

There is a nontrivial relation among the vectors ${\overrightarrow{\mathrm{v}}}_1,{\overrightarrow{\mathrm{v}}}_2,...,{\overrightarrow{\mathrm{v}}}_{\mathrm{m}}$if (and only if) at least one of the vectors${\overrightarrow{\mathrm{v}}}_{\mathrm{i}}$ is a linear combination of the other vectors${\overrightarrow{\mathrm{v}}}_1,{\overrightarrow{\mathrm{v}}}_2,....{\overrightarrow{\mathrm{v}}}_{\mathrm{i}-1},{\overrightarrow{\mathrm{v}}}_{\mathrm{i}+1},....,{\overrightarrow{\mathrm{v}}}_{\mathrm{m}}.$ .

Step-by-step solution

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

$c_{i}v{⃗}_i=-c_{1}v{⃗}_1-c_{2}v{⃗}_2-...-c_{i-1}v{⃗}_{i-1}-c_{i-1}v{⃗}_{i+1}-...-c_{m}v{⃗}_m$

$\Rightarrow {\overrightarrow{v}}_i=-c_i^{-1}{c}_1{\overrightarrow{v}}_1-c_i^{-1}{c}_2{\overrightarrow{v}}_2-.....-c_i^{-1}{c}_{i-1}{\overrightarrow{v}}_{i-1}-c_i^{-1}{c}_{i+1}{\overrightarrow{v}}_{i+1}-...-c_i^{-1}{c}_m{\overrightarrow{v}}_m$

The vector is a linear combination of the other vector .

  To prove the converse part of the given statement

Conversely, let us consider that the vectoris a linear combination of the other vectors${\overrightarrow{\mathrm{v}}}_1,{\overrightarrow{\mathrm{v}}}_2,....{\overrightarrow{\mathrm{v}}}_{\mathrm{i}-1},{\overrightarrow{\mathrm{v}}}_{\mathrm{i}+1},....,{\overrightarrow{\mathrm{v}}}_{\mathrm{m}}.$. Then we can write

${\overrightarrow{\mathrm{v}}}_1=d_1{\overrightarrow{\mathrm{v}}}_1+d_2{\overrightarrow{\mathrm{v}}}_2+...+d_{i-1}{\overrightarrow{\mathrm{v}}}_{i-1}+d_{i+1}{\overrightarrow{\mathrm{v}}}_{i+1}+...+d_m{\overrightarrow{\mathrm{v}}}_m$

Subtractingrole="math" localid="1659359195199" ${\overrightarrow{\mathrm{v}}}_i$from both sides of above equation, we get

role="math" localid="1659359178216" $0=d_1{\overrightarrow{\mathrm{v}}}_1+d_2{\overrightarrow{\mathrm{v}}}_2+...+d_{i-1}{\overrightarrow{\mathrm{v}}}_{i-1}+d_{i+1}{\overrightarrow{\mathrm{v}}}_{i+1}+...+d_m{\overrightarrow{\mathrm{v}}}_m$

Thus, we get a non-trivial relation among the vectors${\overrightarrow{\mathrm{v}}}_1,{\overrightarrow{\mathrm{v}}}_2,...,{\overrightarrow{\mathrm{v}}}_{\mathrm{m}}.$ if the vector is a linear combination of the other vectors${\overrightarrow{\mathrm{v}}}_1,{\overrightarrow{\mathrm{v}}}_2,....{\overrightarrow{\mathrm{v}}}_{\mathrm{i}-1},{\overrightarrow{\mathrm{v}}}_{\mathrm{i}+1},....,{\overrightarrow{\mathrm{v}}}_{\mathrm{m}}.$ .

  Final Answer

There is a nontrivial relation among the vectors ${\overrightarrow{\mathrm{v}}}_1,{\overrightarrow{\mathrm{v}}}_2,...,{\overrightarrow{\mathrm{v}}}_{\mathrm{m}}.$if (and only if) at least one of the vectors${\overrightarrow{\mathrm{v}}}_i$ is a linear combination of the other vectors${\overrightarrow{\mathrm{v}}}_1,{\overrightarrow{\mathrm{v}}}_2,....{\overrightarrow{\mathrm{v}}}_{\mathrm{i}-1},{\overrightarrow{\mathrm{v}}}_{\mathrm{i}+1},....,{\overrightarrow{\mathrm{v}}}_{\mathrm{m}}.$ .