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This may be a redundant question because I'm still studying this field. I'm attempting to write an algorithm that removes linearly dependent vectors from a matrix in a simple way.

First approach was to rewrite the definition $c_1v_1 + \cdots + c_nv_n = 0$ as

$ \begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix} \begin{pmatrix} c_1 & \cdots & c_n \end{pmatrix} = 0 $

and solve for the constants, not sure if this is an optimal or even correct approach but I can't seem to reach some identification of a linearly dependent vector.

how can one approach this correctly?

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  • $\begingroup$ I think you want an algorithm, not a function. $\endgroup$ – Riquelme Aug 31 '19 at 21:05
  • $\begingroup$ @Riquelme Correct. I meant a programming function and not a mathematical function, I'm used to computer science references. $\endgroup$ – Tomergt45 Aug 31 '19 at 21:09
  • $\begingroup$ The method you meant is valid see: math.oregonstate.edu/home/programs/undergrad/…. Another approach if dividing each row by each other row and each column by each column yield save value respectively. The later is easier to program. $\endgroup$ – NoChance Aug 31 '19 at 21:46
  • $\begingroup$ The QR decomposition mentioned in an answer below is probably the most computationally-efficient method, but you can also do this with good old Gaussian elimination or the Gram-Schmidt process. $\endgroup$ – amd Aug 31 '19 at 22:01
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You can do QR decomposition programmatically (e.g., using numpy.linalg.qr), then every $v_i$ corresponding zero diagonal component of $R$ are what you're supposed to remove!

That is, once you get the upper triangular matrix, $R$, every $v_i$ for which $R_{ii}=0$ is the vector linearly dependent on $v_1, \ldots, v_{i-1}$.

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