Determining Linear dependency in a system of trilinear equations I am currently studying linear algebra, and I'm having a hard time understanding the concept of linear dependency...
I understand the fact that in order to determine if two vectors are linearly dependent, they need to be multiples of one another.
This has lead me to the following conclusions, however I'm not too sure if they're right:

*

*In a system of linear equations, if you have infinite solutions, system is dependent

*In a system of linear equations, if you have one solution, system is independent

*In a system of linear equations, if you have no solution, system is inconsistent

*In a system of linear equations, if the answer is null vector, system is independent
Note: I apologize if the terms are misspelled or they are named a little bit different in English. I am currently studying math in another language, and I'm assuming these are the correct translations.
 A: I would first define the linear dependency in terms of vectors, and that helps to conclude what it gives us in regards to linear equations.

I understand the fact that in order to determine if two vectors are linearly dependent, they need to be multiples of one another.

Kind of. The linear dependency is defined for set of some arbitrary vectors $\{\vec{v_1}, \vec{v_2} \dots \vec{v_n}\}, \vec{v_i} \in\mathbb{R}^{m} $ where these vectors are linearly dependent if and only if any single of those vectors can be defined by linear combination of the other vectors. Otherwise they are linearly independent.

*

*So linearly independent set of vectors might look like this:
$\left\{\vec{v_1}=\begin{bmatrix}9\\0\end{bmatrix},\vec{v_2}=\begin{bmatrix}0\\-4\end{bmatrix}\right\}$,
since it's impossible to define $\vec{v_2}$ with any set of linear
combination (in this case it's narrowed down to scalar
multiplication) of $\vec{v_1}$. And there is
exactly one point on the coordinate plane where multiples of those
vectors can intersect each other (since they are not collinear).


*If we take other vectors in the set
$\left\{\vec{v_1}=\begin{bmatrix}1\\2\end{bmatrix},\vec{v_2}=\begin{bmatrix}2\\4\end{bmatrix}\right\}$
they are linearly dependent because $\vec{v_2}=2\vec{v_1}$. Such
vectors are collinear.


*Another sample of linearly dependent set of vectors is essentially
any set of vectors $\in \mathbb{R}^m$ which contains more than $m$
elements even if they are not collinear. I'm not sure if there is such rule, but conventional wisdom says that for any set of linearly independent vectors $\alpha=\{\vec{v_1},\vec{v_2}\dots\vec{v_m}\}, \vec{v_i}\in\mathbb{R}^m$ the span would be $span(\alpha)=\mathbb{R}^m$. E.g. with linear combinations of the vectors in the set of the first example $\left\{\begin{bmatrix}9\\0\end{bmatrix},\begin{bmatrix}0\\-4\end{bmatrix}\right\}$ you can define any other vector in $\mathbb{R}^2$ space, and by definition it means that any third vector of the same space makes this set linearly dependent (say, set $\left\{\begin{bmatrix}9\\0\end{bmatrix},\begin{bmatrix}0\\-4\end{bmatrix},\begin{bmatrix}1\\1\end{bmatrix}\right\}$ is linearly dependent one way or another).
If we consider the following system of a linear equations, say, with two unknowns $\begin{equation}\begin{cases}ax+by=p_1\\cx+dy=p_2\end{cases}\end{equation}$ with matrix of coefficients $\begin{bmatrix}a & b\\c & d\end{bmatrix}$ we can use the set of vectors from this matrix to make the following conclusion about the number of solutions:

*

*If $\{\begin{bmatrix}a\\c\end{bmatrix}\,\begin{bmatrix}b\\d\end{bmatrix}\}$ is linearly independent, there is exactly one solution (and the null vector is just one of possible solutions to such a system).

*If $\{\begin{bmatrix}a\\c\end{bmatrix}\,\begin{bmatrix}b\\d\end{bmatrix}\}$ is linearly dependent, the system is either inconsistent or it has infinite numbers of solutions.

The linear dependency of the third scenario actually comes down to a underetermined system, which by definition also has infinite or zero solutions.
Edit: It's important to note, however, that these tricks are applicable to the vectors made of coefficients only. In terms of systems of equations, where we must use an augmented matrix, the inconsistent cases are considered linearly independent system of equations, because they yeld different graphs.
