Question: Determine if
$\vec v_1=(3,8,7,-3)$, $\vec v_2=(1,5,3,-1)$, $\vec v_3=(2,-1,2,6)$, $\vec v_4=(1,4,0,3)$
are linearly independent.
I know how to do this question but I do not fully understand what it means when vectors are linearly independent.
The definition of linear independence says that:
- A non-empty set of vectors $S$ is linearly independent if the scalars in the linear combination of all vectors is zero, $c_1,c_2,...,c_k=0$ for $$c_1\vec v_1+c_2\vec v_2+ ... + c_k\vec v_k = \vec 0$$
- A non-empty set of vectors $S$ is linearly dependent if the scalars in the linear combination of all vectors is not all zero, for $$c_1\vec v_1+c_2\vec v_2+ ... + c_k\vec v_k = \vec 0$$
I find this definition not useful. My understanding of linear independence is that given a nonempty set of vectors, the set is linearly independent if all given vectors point in different directions and the only common direction (vector) is the origin (zero vector), and it is linearly dependent if all given vectors point in the same direction.
I am wondering what linear independence actually means geometrically and if my explanation above is correct.
And also for the original question, rather than setting up an augmented matrix and showing that the only solution is the trivial solution for linear independence, why can I not do the following instead to show linear independence/dependence?
- Find the unit vectors of $\vec v_1$ $\vec v_2$ $\vec v_3$ $\vec v_4$ (so all have the length 1 and only show the direction)
- If the unit vectors are all the same, it means the four vectors point in the same direction and so they are linearly dependent.
- If the unit vectors are all different, it means the four vectors point in different directions and so they are linearly independent.