Question

Show that the set of vectors {(1,1,5),(1,1,3)(1,1,3)}

it is linear independent or not

Answer 1

Solution;

A set of vectors,v_1,v₂,v₃ are linearly independent if the only scalars that satisfy;

k₁v₁+k₂v₂+k₃v₃=0.....(1)

Are;

k₁=k₂=k₃=0.

The equivalent homogeneous solution of (1) is;

$\begin{bmatrix} | & |&| \\ v_1& v_2&v_3\\ |&|&| \end{bmatrix}$ $\begin{bmatrix} k_1 \\ k_2\\ k_3 \end{bmatrix}=0$

The coefficient vector matrix is;

$\begin{bmatrix} 1 & 1&1\\ 1 & 1&1\\ 5&3&3 \end{bmatrix}$

The reduced row echelon matrix form is;

$\begin{bmatrix} 1 & 0&0 \\ 0 & 1&1\\ 0&0&0 \end{bmatrix}$

This shows that there exists a nontrivial linear combination of the vectors. Hence the set of vectors is linearly dependent.