is a subspace of if
1) for some vectors .
2) satisfies the closure properties:
(i) for all , we must have .
(ii) for all and , we must have .
3) is the solution set of a homogeneous system.
(Sufficient to check either one of Condition 1, 2, 3.)
For to be a subspace, zero vector must be in . (Since for , , we have .)
Linear Independence and Dependence
are linearly independent if the system has only the trivial solution, i.e. .
If the system has non-trivial solutions, i.e. at least one not zero, then are linearly dependent.