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.
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