I've come across this word a few times. Although I eventually figure out what it means, I can never remember it. Can someone explain it to me and maybe show an example, please?
I believe that this involves free variable. I think linear dependency may not have any free variables or at least not everything is a free variable. I do know that linear independent has only the trivial solution for an equation.
Am I at least understand some of what linear dependent is?
On
If you look at the definition it says a set of vectors $v_1, v_2, v_3, . . . v_m$ is said to be linearly dependent if it's possible to find scalars $\alpha_1, \alpha_2, \alpha_3, . . . \alpha_m$ not all zero such that
$\alpha_1v_1+\alpha_2v_2+\alpha_3v_3+...+\alpha_mv_m=0$ m
Now observe the fact that not all scalars are zero; to understand it in a better way assume $\alpha_1$ is not zero .
Then we can write above as
$v_1=-\alpha_{1}^{-1}[\alpha_2v_2+\alpha_3v_3+...+\alpha_mv_m]$
Means $v_1$ is not a new vector but is some linear combination of $v_2, v_3, ..., v_m$
In other words $v_2, v_3, ..., v_m$ can be used to generate $v_1$
That means there is some sort of dependency in the set $v_1, v_2, v_3, ..., v_m$
Hence the name linearly dependent
A set of vectors is linearly dependent if any one of them is a linear combination of others. For example, $(1,2),(2,4)$ are linearly dependent because $(2,4)=2(1,2)$. Whereas $(1,2),(2,3)$ are linearly independent.
The determinant of linearly dependent (independent) vectors is $0$ (nonzero). For above examples: $$\begin{vmatrix} 1 & 2 \\ 2 & 4 \end{vmatrix}=0.$$ $$\begin{vmatrix} 1 & 2 \\ 2 & 3 \end{vmatrix} =-1 \ne 0.$$