I have a set of vectors. For simplicity, suppose there are $n=20$ vectors and each has $p=5$ elements. These vectors are generated from some experiments, and so it is not very apparent how they are linearly dependent. A vector can be $V_1=[1,0,2,4,2.5]$ and another may be $V_2=[1.9,0,4.05,8,5.01]$, which seems like they are not linearly dependent, but if we look closely, it seems that the second vector could be a noisy version of the vector $V_2'=[2,0,4,8,5]$ which is definitely related to $V_1$.
So, if I have a situation like this, that may be out of the 20 vectors 6 are linearly related to each other but are noisy, another 12 are also linearly related together and the rest of the 2 are completely different. So, basically it seems that this set of 20 vectors can be represented by just 4 vectors, the 1st one represents the set of 6, the 2nd one represents the set of 12 and the last two are the two unique ones. How do I find these 4 vectors?
Also, I don't want to reduce the dimension of the vectors, they should have $p=5$ elements. I just want to reduce the number of vectors in the set and remove the redundancy.
I looked into Principal Components Analysis (PCA) but PCA reduces the dimension of the vector, not the redundancy which I don't want.
Also, the groups of 6 and 12 are just for example, I will not know in advance how many vectors in the set of 20 are similar to each other. Can anyone help me out with this problem?
Thanks in advance.