Can the determinant (assuming it's non-zero) be used to determine that the vectors given are linearly independent, span the subspace and are a basis of that subspace? (In other words assuming I have a set which I can make into a square matrix, can I use the determinant to determine these three properties?)
Here are two examples:
- Span Does the following set of vectors span $\mathbb R^4$: $[1,1,0,0],[1,2,-1,1],[0,0,1,1],[2,1,2,-1]$? Now the determinant here is $1$, so the set of vectors span $\mathbb R^4$.
- Linear Independence Given the following augmented matrix:
$$\left[\begin{array}{ccc|c} 1 & 2 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \end{array}\right], $$ where again the determinant is non-zero ($-2$) so this set S is linearly independent.
Of course I am in trouble if you can't make a square matrix - I figure for spans you can just rref it, and I suppose so for linear independence and basis?