Dot Product
–
–
Span
Subspaces
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.)
Remark:
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.