How to find linearly dependent subsets from a large set of real-valued vectors quickly? Problem
I have a set $V \subset \mathbb{R}^n$, where $n = 600$.
The set of vectors $V$ is big enough: $\left| V \right| \approx 50 \cdot 10^3$.
I want to find $m$-tuples $U_i \subset V$ with linearly dependent vectors
\begin{equation}
\begin{cases}
\forall U_i: \exists \alpha^i \neq 0 \in \mathbb{R}^n: \sum\limits_{u \in U_i} u \cdot \alpha^i_u = 0 \\
\left| U_1 \right| = \left| U_2 \right| = \dots = \left| U_m \right|
\end{cases}
\end{equation}
At the moment I've found out manually that I have no linearly dependent pairs, using orthogonal projection and brute force.
For each pair $\left\langle v, u \right\rangle \in V^2$ (pairs mean $\left| U_i \right| = 2$)
I've checked
\begin{equation}
v - \frac{\left( v, u \right)}{\left\| u \right\|^2} \cdot u = 0
\end{equation}
and it was false for all pairs.
Also this consumed hours 20 minutes on my laptop: I've needed to check about
$\frac{\left( 50 \cdot 10^3 \right)^2}{2} = 1.25 \cdot 10^9$ pairs.
Similar approach looks impossible for the case of triples ($1.25 \cdot 10^{14}$ items), not saying about quads ($6.25 \cdot 10^{18}$ tuples).
Details
There were 199 tuples with $50 \cdot 10^3$ 3D vertices each (199 3D scans of faces), which were analyzed with PCA in order to have a generative face model.
As result I've got set of $50 \cdot 10^3$ vectors.
Each vector contains average position and principal components coefficients of corresponding vertex.
I'm working on method, which uses this information, and it needs triples of vertices, in which offset coefficients are linearly dependent and average positions are not.
This means, that I have $n = 597$, not $600$, but does it mean for an algorithm?
Sure, I can use not all coefficients, for example $50$ from each dimension, which will give me $\mathbb{R}^{150}$.
Though, I'm working on generic approach and in another case number of dimensions can be bigger.
Question
Is there a fast algorithm for finding at least all triples and quads of linearly dependent vectors from a given set in multidimensional space (hundreds of dimensions)?
Obviously, storage of an ordinary computer cannot store all possible tuples and algorithm has to give solution by chunks.
 A: Since my comment is a bit too long, I will write that as an answer, and delete it afterwards.
Maybe it helps you, maybe it is completely wrong.
I understand your question in the following way: 
Given a set of vectors $V$ find all possible m-tuple $U⊂V$, with $2≤m≤600$, of linear dependant vectors. 
Here are two possible optimization to the brute force algorithm: 


*

*Linear dependence [l.d.] of two vectors is transitive: 
If $v_1$ and $v_2$ are l.d. and $v_2$ and $v_3$ are l.d. than  $v_1$ and $v_3$ are l.d. 
⇒ You can build equivalence classes of l.d. vectors. 
⇒ Here with only two vectors ($v_1, v_2$) in your equivalence class, you save 1 calculation. If you already now your equivalence class contains $m$ vectors, you save $m-1$ calculations.
I am currently not sure if that can be generalised to sets of vectors.

*If you want to know if $2$ vectors $u,v∈ℝ^n$ are linear independant, it is enough to check if the following $2×2$-matrix $$A=\begin{pmatrix} u_1 & v_1 \\ u_2 & v_2 \end{pmatrix}$$ with the first two components, has full rank (=is invertible). 


*

*If yes, then $u$ and $v$ are linear independant, and thus not l.d. 

*If no, you can't say anything. 

*$m$ vectors ⇒ $m×m$ matrix
Does this help?
edit: I just had a thought about your check:
$$v−\frac{(v,u)}{∥u∥^2}\cdot u==0$$
Are your vectors from experimental / computational data? If so this will never ever be ==0. Rather check $≤10^{−8}$.
