How to Determine if a Set is a Vector Space

When you want to determine if a set is a vector space, you have to verify that the set, along with the defined linear operations of vector addition and scalar multiplication, satisfies a specific set of axioms. This article will guide you through the process of checking if a given set qualifies as a vector space. We will break down the necessary conditions and provide an example of how to apply these conditions.

What is a Vector Space?

A vector space is a mathematical structure consisting of a set of vectors, the operations of vector addition, and scalar multiplication. The set, together with these operations, must adhere to a set of axioms, known as vector space axioms, to be considered a vector space.

Vector Space Axioms

To determine if a set is a vector space, you must check that the set, along with the defined operations of vector addition and scalar multiplication, satisfies the following ten axioms:

Axiom L1: Closure under Addition

For all vectors (x, y in V), the sum (x y) is also in (V).

Axiom L2: Commutative Property

For all vectors (x, y, z in V), (x (y z) (x y) z).

Axiom L3: Identity Element

There exists a zero vector (0 in V) such that for all vectors (x in V), (x 0 0 x x).

Axiom L4: Inverse Element

For each vector (x in V), there exists a vector (-x in V) such that (x (-x) (-x) x 0).

Axiom L5: Associative Property

For all vectors (x, y in V), (x y y x).

Axiom L6: Closure under Scalar Multiplication

For all scalars (lambda in F) and vectors (x in V), the product (lambda x) is also in (V).

Axiom L7: Identity Scalar Element

For all vectors (x in V), (1x x), where (1 in F).

Axiom L8: Distributive Property over Scalars

For all scalars (lambda, mu in F) and vectors (x in V), ((lambda mu)x lambda x mu x).

Axiom L9: Distributive Property over Vectors

For all scalars (lambda in F) and vectors (x, y in V), (lambda(x y) lambda x lambda y).

Axiom L10: Associative Property of Scalar Multiplication

For all scalars (lambda, mu in F) and vectors (x in V), ((lambda mu)x lambda(mu x)).

Example: Checking If a Set is a Vector Space

Let's consider the set of all polynomials of degree less than or equal to (n), denoted as (P_n), and show that it forms a vector space.

1. **Closure under Addition:** The sum of two polynomials is still a polynomial of degree less than or equal to (n).

2. **Closure under Scalar Multiplication:** Any scalar multiple of a polynomial is still a polynomial of degree less than or equal to (n).

3. **Additive Identity:** The zero polynomial (0) is included.

4. **Additive Inverse:** For each polynomial (p(x)), (-p(x)) is still a polynomial of degree less than or equal to (n).

5. **Commutativity and Associativity of Addition:

6. **Identity of Scalar Multiplication:

7. **Distributive Property of Scalars:

8. **Distributive Property of Vectors:

9. **Associativity of Scalar Multiplication:

Since all ten axioms are satisfied, the set (P_n) with these operations forms a vector space.

Conclusion

In summary, determining if a set is a vector space involves checking that the set, along with the defined linear operations, satisfies the ten axioms presented. This process requires careful verification of each axiom to ensure that the set qualifies as a vector space.

To learn more about vector spaces, consider exploring the resources at Kip Ingrams Space.