Linear Independence
Vectors $\mathbf x_1, \mathbf x_2, ... , \mathbf x_n$ are linearly independent if no linear combinations gives a zero vector $\mathbf 0$
 $c_1 \mathbf x_1 + c_2 \mathbf x_2 + ... + c_n \mathbf x_n \ne \mathbf 0$
 only when $\forall i: c_i = 0$ it's true that $\sum c_i \mathbf x_i = 0$
Examples
Example 1
 suppose we have a vector $v_1$ and a vector $v_2 = c \cdot v_1$

 this system is dependent: $v_2$ points in the same direction as $v_1$
Zero vector always means a dependence
 suppose we have vector $v_1 \ne \mathbf 0$ and $v_2 = \mathbf 0$
 $0 \mathbf v_1 + c \mathbf v_2 = \mathbf 0$ for any $c$
Example 2
 suppose we have a system of two independent vectors $v_1$ and $v_2$ in $\mathbb R^2$

 what happens if we add 3rd vectors $v_3$?

 the system is no longer dependent  we always can express $v_3$ in terms of $v_1$ and $v_2$ and when we add them, we'll have 0
 so if the number of vectors is greater than the dimensionality of these vectors, the system cannot be independent
Matrices
Columns of a matrix $A$ are independent if the Nullspace $N(A)$ contains only $\mathbf 0$
 otherwise the columns are dependents
 Why? recall that $N(A)$ contains the solutions to the system $A\mathbf x = \mathbf 0$
 so there's a combination of columns with coefficients $\mathbf x$ that is equal to $\mathbf 0$
 It's related to rank as well:
 if $r = n$, then there are no free variables and $N(A) = \{ \, \mathbf 0 \, \}$
 if $r < n$, there are free variables and $ N(A)  > 1$
Maximal Linearly Independent Subset
Suppose we have a set of vectors $A = \{ \mathbf a_i \}$
 subset $A^*$ is maximal linearly independent subset of $A$ if
 all vectors in $A^*$ are linearly independent
 it's not contained in any other subset of linearly idependent vectors from $A$
Sources