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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 ) ?

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It is possible to have a set of $n$ linearly dependent vectors such that all nonempty proper subsets are linearly independent.

For example, in $\mathbb{R}^{n-1}$, take the standard basis vectors $e_1,\ldots,e_{n-1}$ along with the vector $e_1+\ldots+e_{n-1}$. All together, the vectors form a linearly dependent set, but any $n-1$ of them are linearly independent.

So to check for linear independence, you cannot do so by 'mini-checks'.

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