Question

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

Short answer

If there are some linearly independent vectors in and a vector ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,..{\overrightarrow{v}}_m,$in ${\mathrm{ℝ}}^{n}andavector\overrightarrow{v}in{\mathrm{ℝ}}^n$that is not contained in the span of${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,...{\overrightarrow{v}}_m$ then the vectors are necessarily linearly independent.

Step-by-step solution

  Linear Relations

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

${\mathit{a}}_{\mathbf{1}}{\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{+}{\mathit{a}}_{\mathbf{2}}{\overrightarrow{\mathbf{v}}}_{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathit{a}}_{\mathbf{m}}{\overrightarrow{\mathbf{v}}}_{\mathbf{m}}\mathbf{=}\mathbf{0}$ (1)

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

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

Since the vector$\overrightarrow{v}$in ${\mathrm{ℝ}}^n$is not contained in the span of ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,...{\overrightarrow{v}}_m,$. Therefore

$\overrightarrow{v}\ne a_1{\overrightarrow{v}}_1,+a_2{\overrightarrow{v}}_2+...+a_m{\overrightarrow{v}}_m$, for$a_i\ne 0$ . (2)

Now multiply the equation (2) byrole="math" localid="1659416449027" $b\in \mathrm{ℝ}andb\ne 0$from left ,then we have

$b\overrightarrow{v}\ne b{a}_1{\overrightarrow{v}}_1+b{a}_2{\overrightarrow{v}}_2+...+b{a}_m{\overrightarrow{v}}_m$ (3)

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

role="math" localid="1659416709972" $0\ne b{a}_1{\overrightarrow{v}}_1+b{a}_2{\overrightarrow{v}}_2+...b{a}_m{\overrightarrow{v}}_m-b\overrightarrow{v}$ (4)

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

Hence, the vectors ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,...{\overrightarrow{v}}_m{\overrightarrow{v}}_{}$are linearly independent.

Final Answer

If there are some linearly independent vectors in ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,...{\overrightarrow{v}}_{m}in{\mathrm{ℝ}}^n$and a vector ${\mathrm{ℝ}}^n$in that is not contained in the span${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,...{\overrightarrow{v}}_m$ of then the vectors ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,...{\overrightarrow{v}}_m,\overrightarrow{v}$are necessarily linearly independent.