Given ${N-1}$ vectors $v_1, v_2, ... ,v_{N-1}$ in which span vector space $V \subset \mathbb{R}^M$, with $M>N$, is there any easy way to check if a given vector $w$ is contained in $V$? If $V \subset \mathbb{R}^N$, I can just find the determinant of a matrix $M = [V \space w]$
Would $det(M' M)$ suffice?
EDIT: This isn't homework; I'm just curious if there's any simple/neat ways I'm not thinking of to check this.
On
Quite simply: consider the matrix $A$ with column vectors the coordinates of $v_1, \dots, v_{N-1}$ and the augmented matrix $A'=Aw$ with the coordinates of $w$ as the last column. The $w\in V$ if and only if $$\operatorname{rank} A=\operatorname{rank} A'.$$ This can be easily checked by row reduction, and it even allows you to find the coefficients of a linear combination of some of the $v_i$s which is equal to $w$.
When actually figuring out if a set of vectors is Linearly dependent, yes you put them in a matrix and check the determinant, since what you are doing is solving the system $$ a_1v_1+a_2v_2+.....+a_nv_n=0 $$ For $a_i$ not all zero, which can only happen if the matrix made up of the vectors $v_1,...,v_n$ as rows or columns is not injective (nontrivial kernel).