Question

Which one of the following sets of vectors is linearly independent?

{(1,1,0),(1,0,1),(−5,−8,4)}

{(1,1,0),(1,0,1),(4,0,4)}

{(1,1,0),(1,0,1),(2,1,1)}

{(1,1,0),(1,0,1),(−5,−9,4)}

Answer 1

Solution;

A set of vectors,v₁,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;

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

For set 1;

The vector matrix is;

$\begin{bmatrix} 1 & 1&-5\\ 1 & 0&-8\\ 0&1&4 \end{bmatrix}$

Reduced row echelon matrix form is;

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

From the echelon form,it seen that;

k₃=0and it follows that k₂=k₁=0

Hence,the set of vectors are linearly independent.

For set 2;

The vector matrix is;

$\begin{bmatrix} 1 & 1&4 \\ 1& 0&0\\ 0&1&4 \end{bmatrix}$

The row reduced echelon form is;

$\begin{bmatrix} 1 & 0&0\\ 0 & 1&4\\ 0&0&0 \end{bmatrix}$

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

Fro set 3;

The vector matrix is;

$\begin{bmatrix} 1 & 1&2\\ 1 & 0&1\\ 0&1&1 \end{bmatrix}$

The reduced row echelon form is;

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

Hence the vectors are linearly dependent.

For set 3;

The vector matrix is;

$\begin{bmatrix} 1 & 1&-5 \\ 1&0& -9\\ 0&1&4 \end{bmatrix}$

The row reduced echelon form is;

$\begin{bmatrix} 1 & 0&-9 \\ 0& 1&4\\ 0&0&0 \end{bmatrix}$

Hence the vectors are linearly dependent.

Answer;

{(1,1,0),(1,0,1),(-5,-8,4)} is linearly independent.