Independent set of vectors in a dependent set. If I have a dependent set of vectors, how do I know which of them are linearly independent between them, and which vectors exactly are dependent to the others?
For example: If I have 5×5 matrix of rank 4. Does that mean that if I take any 4 vectors out of the matrix they are linearly independent?If not, how do I know which ones are exactly independent and whis is the one dependent?
Thank you in advance and I apologise for not being very clear.
 A: 
If I have a dependent set of vectors, how do I know which of them are linearly independent between them, and which vectors exactly are dependent to the others?

Answer: Gauss's elimination.
Explanation: Suppose that you have the following set $S=\left\{v_{1},v_{2},v_{3},\ldots,v_{k}\right\}\subset V$ where $V$ is a vector space on field $F$. Assuming that we know that in the set $S$ there is a subset of $S$ that is dependent linearly and there is another subset of $S$ such that is independent linearly, so your question is: how to recognize which of the vectors is linearly dependent and which of the vectors is linearly independent? The answer is behind: Gaussian row reduction (row elimination).
Take the coordinates of the vectors in $S$, and then reduce by rows the matrix that contains the coordinate vectors in its rows, the rows that vanish were associated with the linearly dependent vectors and the rows that did not vanish were associated with the vectors linearly independent.
Example:  Let's to study the subset $$S=\left\{ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix},\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix} \right\}\subset \mathbb{R}^{3}$$
Now, in the set $S$ is clear that $\{(1,0,0),(0,1,0),(0,0,1)\}$ is independent linearly in $\mathbb{R}^{3}$ and $\{(1,1,1),(2,2,2)\}$ is dependent linearly in $\mathbb{R}^{3}$. But let's to use the elimination method for check it:
Let $$A=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1\\ 2 & 2 & 2 \end{pmatrix}$$
for Gauss's elimination we can see that
$$A=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \color{blue}{1} & \color{blue}{1} & \color{blue}{1}\\ \color{red}{2} & \color{red}{2} & \color{red}{2} \end{pmatrix}\sim \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \color{blue}{0} & \color{blue}{0} & \color{blue}{0}\\ \color{red}{0} & \color{red}{0} & \color{red}{0} \end{pmatrix} $$
since that the row $\color{blue}{4}$ and $\color{red}{5}$ is associated for the vectors $(1,1,1)$ and $(2,2,2)$,  then by the Gauss's elimination we can conclude that the set $\{(1,1,1),(2,2,2)\}$ is dependent linearly in $\mathbb{R}^{3}$ and that the set $\{(1,0,0),(0,1,0),(0,0,1)\}$ is independent linearly in $\mathbb{R}^{3}$.
