Suppose i have a set of linearnly independent vectors v1,v2,v3 outside span[b1,b2] and i wanna check that $v_1,v_2,v_3,b_1,b_2$ forms a linearnly independent set.
Can i do this mini-checkups of linearnly independence :
1 - $b_1,b_2,v_1$ ;
2 - $b_1,b_2,v_2$ ;
3- $b_1,b_2,v_3$ ;
4 - $b_1,b_2,v_1,v_2$ ;
5 - $b_1,b_2,v_2,v_3$ ;
to conclude that $v_1,v_2,v_3,b_1,b_2$ are linearly independent ?
Do I also need the last test :
6 - $b_1,b_2,v_1,v_3$ ; ?
Or actually the only way is to append one by one and check if it remains linear independent until I append every vector ?
If I can do the mini-checks, can I generalize to any number of vectors $b_1,b_2,...,b_z$ and $v_1,v_2,...,v_n$ provided I do the necessary mini-checks ( dictated by combinatorics ) ?