how do I check linear independence of m vectors, given the m-1 first vectors are independence?

Question

I have a set of m-1 vectors of size n, they are independence. I want to add another vector and to check if the m vectors are still independence. It is known m < n. I am looking for a good algorithm & running time. Thanks

Answer 1

You can use the following criterion :

If you have $\ m-1\ $ linear independent vectors, then adding the $\ m\ $- th vector gives again a linear independent set if and only if the $\ m\ $ - th vector is not a linear-combination of the original $\ m-1\ $ vectors, in other words there are no real numbers $\ a_1,\cdots, a_{m-1}\ $ with $$\sum_{j=1}^{m-1} a_jv_j=v_m$$

For the check I suggest to collect the vectors in a matrix and determine the rank.

Answer 2

Project the new vector on the others and remove.

If anything remains afterwards then they are linearly independent.