1. Section 4.1: Vector Spaces and Subspaces

2. REVIEW
Recall the following algebraic properties of

3. Definition
A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars, subject to the ten axioms:

4. Examples
= the set of polynomials of degree at most n
: the set of all real-valued functions defined on a set D.

5. Definition
A subspace of a vector space V is a subset H of V that satisfies
The zero vector of V is in H.
H is closed under vector addition.
H is closed under multiplication by scalars.

6. Properties a-c guarantee that a subspace H of V is itself a vector space.
Why? a, b, and c in the defn are precisely axioms 1, 4, and 6. Axioms 2, 3, 7-10 are true in H because they apply to all elements in V, including those in H. Axiom 5 is also true by c.
Thus every subspace is a vector space and conversely, every vector space is a subspace (or itself or possibly something larger).

7. Examples:
The zero space {0}, consisting of only the zero vector in V
is a subspace of V.

8. 5. Given and in a vector space V, let
Show that H is a subspace of V.

9. Theorem 1
If are in a vector space V,
then Span is a subspace of V.

10. Example: