Check whether a set of vectors is independent

Question

Is the following set of vectors independent or dependent:

$$(1,-2,5,-3)^T, \;\;(2,3,1,-4)^T \;\; and \;\; (3,8,-3,5)^T \;\;?$$

How do I determine whether this set is dependent or independent ?

Answer 1

Let $v_1, v_2, v_3$ be the three vectors you listed above.

What you need to do is to solve the equation $c_1v_1+c_2v_2+c_3v_3=0$.

If $c_1=c_2=c_3=0$ is the only solution, then they are independent. If there are nontrivial solutions, then they are dependent.

This is the basic definition of linearly independent.

Regarding the solving part, if you have a scientific calculator that can solve simultaneous equations, you are good to go. Otherwise, you would need to use Gaussian Elimination.

Answer 2

For $A = (a_1, a_2, a_3)$ and $c = (c_1, c_2, c_3)^\top$ you want to solve $$ A c = 0 $$ and find out if only the trivial solution exists. You can find out by Gauss elimination on the augmented matrix (you can also drop the augmented column, as it stays zero): \begin{align} \left[ A \mid 0 \right] &= \left[ \begin{array}{rrr|r} 1 & 2 & 3 & 0 \\ -2 & 3 & 8 & 0 \\ 5 & 1 & -3 & 0 \\ -3 & -4 & 5 & 0 \end{array} \right] \to \left[ \begin{array}{rrr|r} 1 & 2 & 3 & 0 \\ 0 & 7 & 14 & 0 \\ 0 & -9 & -18 & 0 \\ 0 & 2 & 14 & 0 \end{array} \right] \to \left[ \begin{array}{rrr|r} 1 & 2 & 3 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & -9 & -18 & 0 \\ 0 & 2 & 14 & 0 \end{array} \right] \\ & \to \left[ \begin{array}{rrr|r} 1 & 2 & 3 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 27 & 0 \\ 0 & 0 & 4 & 0 \end{array} \right] \to \left[ \begin{array}{rrr|r} 1 & 2 & 3 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 4 & 0 \end{array} \right] \to \left[ \begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \end{align} So we have the only solution $c = (0,0,0)^\top$, the vectors are linear independent.