Create 2 sets, one linearly independent and one linearly dependent, by replacing the x with a number, starting from B = { (1,2,-1,8), (-2,-1,4,2), (13,14,-21,x) }

Step: 1

Linearly Dependent: - A set of vectors is called linearly dependent if at least one element of the set can be written as linear combination of others elements.

 Linearly Dependent: - If no vector (element) of set can be written as linear combination of others elements. Then set is called linearly dependent.

Step: 2

Now, given that $B=\left\{\right(1, 2, -1, 8), (-2, -1, 4, 2), (13, 14, -21, x\left)\right\}$

Let B is linearly dependent set then at least one element of B can be written as linear combination of others element.

$$\begin{gathered} So, \\ (13, 14, -21, x)=a.(1, 2, -1, 8) + b.(-2, -1, 4, 2) \\ Here, a and b are scalar. \end{gathered}$$

comparing both sides we get,

$$\begin{gathered} a-2b=13-- \left(1\right) \\ 2a-b=14--\left(2\right) \\ -a+4b=-21--\left(3\right) \\ -a+2b=x--\left(4\right) \\ solving equationn \left(1\right) and \left(2\right), to find value of a and b \\ from \left(1\right), \\ a=13+2b \\ put in \left(2\right), \\ 2a-b=14 \\ 2(13+2b) -b =14 \\ 26+4b-b=14 \\ 3b=-12 \\ b=-4 \end{gathered}$$

$$\begin{gathered} we have, \\ a=13+2b \\ a=13+2\times (-4) \\ a=13-8 \\ a=5 \end{gathered}$$