Question

For which values of the constants$\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{m}$ are the given vectors linearly independent?

$\mathbf{\left[}\begin{array}{c}\mathbf{a}\\ \mathbf{b}\\ \mathbf{c}\\ \mathbf{d}\\ \mathbf{1}\\ \mathbf{0}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{c}\mathbf{e}\\ \mathbf{1}\\ \mathbf{0}\\ \mathbf{0}\\ \mathbf{0}\\ \mathbf{0}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{c}\mathbf{f}\\ \mathbf{g}\\ \mathbf{h}\\ \mathbf{i}\\ \mathbf{j}\\ \mathbf{l}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{c}\mathbf{k}\\ \mathbf{m}\\ \mathbf{1}\\ \mathbf{0}\\ \mathbf{0}\\ \mathbf{0}\end{array}\mathbf{\right]}$

Short answer

For any values of $a,b,.....,m$ the vectors,localid="1664209238077" $\left[\begin{array}{c}a\\ b\\ c\\ d\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}e\\ 1\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}f\\ g\\ h\\ i\\ j\\ l\end{array}\right],\left[\begin{array}{c}k\\ m\\ 1\\ 0\\ 0\\ 0\end{array}\right]$can be linearly independent.

Step-by-step solution

Consider the set

The vectors ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,\dots ,{\overrightarrow{v}}_m$in ${\mathrm{ℝ}}^n$are linearly dependent if and only if there are nontrivial relation among them.

If linearly dependent, then, the vectors are said to be redundant.

Consider the vector role="math" localid="1664209158244" $\left[\begin{array}{c}a\\ b\\ c\\ d\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}e\\ 1\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}f\\ g\\ h\\ i\\ j\\ l\end{array}\right],\left[\begin{array}{c}k\\ m\\ 1\\ 0\\ 0\\ 0\end{array}\right]$

Consider the condition

For the vectors, ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3,{\overrightarrow{v}}_4$

$c_1{\overrightarrow{v}}_1+c_2{\overrightarrow{v}}_2+c_3{\overrightarrow{v}}_3+c_4{\overrightarrow{v}}_4=0$

The solution will be trivial for any values of $a,b,.....,m$.

Final answer

For any values of $a,b,.....,m$the vectors, $\left[\begin{array}{c}a\\ b\\ c\\ d\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}e\\ 1\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}f\\ g\\ h\\ i\\ j\\ l\end{array}\right],\left[\begin{array}{c}k\\ m\\ 1\\ 0\\ 0\\ 0\end{array}\right]$can be linearly independent.