Section 4.1: Vector Spaces and Subspaces

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 additionand 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: