Ok, so first I want to verify that $m$ vectors of length $n>m$ are linearly independent. I think I can do this by setting up a matrix and get the rank of this matrix by gauss-elimination.
However afterwards I want to get another vector that is linearly independent to the set I already have. Is there any easy fast way to get this? Maybe with the work I already did (gauss-elimination)?
I think I can just take a standard-basis vector where I have all zeroes after the gauss-elimination. But I'm not sure about this.
We can proceed arranging the $m$ given vectors by rows and in the RREF matrix selecting any column index $i$ without pivot, then we can add the linearly independent vector $e_i$ to the set.