1. Subspaces and Spanning Sets
Definition : Subspaces
For any vector space, a subspace is a subset that is itself a vector space, under the inherited operations.
Note: A subset of a vector space is a subspace iff it is closed under  & .
→ It must contain 0.
Example : Plane in R3
is a subspace of R3.
Proof:
Let → 
with →

2. Example : The x-axis in Rn is a subspace.
Proof follows directly from the fact that
Example :
{ 0 } is a trivial subspace of Rn.
Rn is a subspace of Rn.
Both are improper subspaces.
All other subspaces are proper.
Example : Subspace is only defined wrt inherited operations.
({1}, ; R) is a vector space if we define 11 = 1 and a1=1 aR.
However, neither ({1}, ; R) nor ({1},+ ; R) is a subspace of the vector space (R,+ ; R).

3. Example : Parametrization of a Plane in R3
is a 2-D subspace of R3.
i.e., S is the set of all linear combinations of 2 vectors (2,1,0)T, & (1,0,1)T.
Example : Parametrization of a Matrix Subspace.
is a subspace of the space of 22 matrices.

4. Definition : Span
Let S = { s1 , …, sn | sk  ( V,+,R ) } be a set of n vectors in vector space V.
The span of S is the set of all linear combinations of the vectors in S, i.e.,
with
Lemma : The span of any subset of a vector space is a subspace.
Proof:
Let S = { s1 , …, sn | sk  ( V,+,R ) }
and 
QED
Converse: Any vector subspace is the span of a subset of its members.
Also: span S is the smallest vector space containing all members of S.

5. Example :
For any vV, span{v} = { a v | a R } is a 1-D subspace.
Example :
Proof:
The problem is tantamount to showing that for all x, y R,  unique a,b R s.t.
i.e.,
has a unique solution for arbitrary x & y.
Since
QED

6. Example : P2
Let
Question:
= subspace of P2 ?
Answer is yes since
and
Lesson: A vector space can be spanned by different sets of vectors.
Definition: Completeness
A subset S of a vector space V is complete if span S = V.

7. Example 2.19: All Possible Subspaces of R3
Planes thru 0
Lines thru 0

8. Exercises 2.I.2 1. Consider the set under these operations.
(a) Show that it is not a subspace of R3. (Hint. See Example 2.5).
(b) Show that it is a vector space.
( To save time, you need only prove axioms (d) & (j), and closure under all linear combinations of 2 vectors.)
(c) Show that any subspace of R3 must pass thru the origin, and so any subspace of R3 must involve zero in its description.
Does the converse hold?
Does any subset of R3 that contains the origin become a subspace when given the inherited operations?

9. 2. Because ‘span of’ is an operation on sets we naturally consider how it interacts with the usual set operations. Let [S]  Span S.
(a) If S  T are subsets of a vector space, is [S]  [T] ?
Always? Sometimes? Never?
(b) If S, T are subsets of a vector space, is [ S  T ] = [S]  [T] ?
(c) If S, T are subsets of a vector space, is [ S  T ] = [S]  [T] ?
(d) Is the span of the complement equal to the complement of the span?
3. Find a structure that is closed under linear combinations, and yet is not a vector space. (Remark. This is a bit of a trick question.)
Lemma :
Let S be a non-empty subset of a vector space ( V, + ; R ).
W.r.t. the inherited operations, the following statements are equivalent:
S is a subspace of V.
S is closed under all linear combinations of pairs of vectors.
S is closed under arbitrary linear combinations.

10. Vector sub-space