Is the Set V {x1y1 | xy∈R} a Vector Space?

When dealing with sets and their structure in mathematics, particularly in the realm of linear algebra, it is crucial to understand whether a given set can be considered a vector space. A vector space is defined by a series of postulates concerning vector addition and scalar multiplication. In this article, we will explore the set V {x1y1 | xy∈R} and determine if it satisfies the conditions to be a vector space.

Introduction to Vector Spaces

A vector space over a field F consists of a set V, together with two operations: vector addition and scalar multiplication, satisfying certain axioms. These axioms can be understood as the 8 fundamental postulates of a vector space. We will apply these postulates to the set V to determine its vector space properties.

Set V {x1y1 | xy∈R}

The set V is defined as the collection of all ordered pairs of real numbers (x1, y1) where x and y are real numbers. Let's consider two vectors in V, represented as a (x1, y1) and b (z1, t1), and a scalar m. We need to define the operations of vector addition and scalar multiplication and then check the 8 postulates of a vector space.

Defining Operations and Postulates

1. Closure: The operations of addition and scalar multiplication must be closed within the set V.

1.1 Vector Addition: Let's define the addition of two vectors a (x1, y1) and b (z1, t1) as:

a b (x1 z1, y1 t1)

This operation ensures that the result of the addition is also in V, verifying closure under vector addition.

1.2 Scalar Multiplication: Let m be a scalar. Define scalar multiplication as:

m · a m · (x1, y1) (mx1, my1)

This operation confirms that the result of scalar multiplication is also in V, verifying closure under scalar multiplication.

2. Associative: Vector addition must be associative.

(a b) c a (b c)

This property follows from the associative property of real number addition.

3. Identity Element: The set V must have an identity element for addition, denoted as 0 (0, 0).

a 0 a 0 a for all a in V

The identity element ensures that adding 0 to any vector in V does not change the original vector.

4. Inverse Element: Each vector a must have an inverse -a such that:

a (-a) 0 (-a) a

For a (x1, y1), the inverse -a is (-x1, -y1), and it is straightforward to verify:

(x1, y1) (-x1, -y1) (0, 0) (-x1, -y1) (x1, y1)

5. Commutative: Vector addition must be commutative.

a b b a for all a, b in V

This property can be easily verified:

(x1, y1) (z1, t1) (z1, t1) (x1, y1)

6. Distributivity (Over Addition): Scalar multiplication must distribute over vector addition.

m · (a b) m · a m · b for all m in F and a, b in V

This distributivity can be verified as follows:

m · ((x1, y1) (z1, t1)) m · (x1 z1, y1 t1) (m(x1 z1), m(y1 t1))

and

m · (x1, y1) m · (z1, t1) (mx1, my1) (mz1, mt1) (mx1 mz1, my1 mt1) (m(x1 z1), m(y1 t1))

7. Distributivity (Over Scalar Multiplication): Scalar multiplication must distribute over scalar addition.

(m n) · a (m · a) (n · a) for all m, n in F and a in V

This can be verified as follows:

(m n) · (x1, y1) ((m n)x1, (m n)y1) (mx1 nx1, my1 ny1) (mx1, my1) (nx1, ny1) (m · (x1, y1)) (n · (x1, y1))

8. Identity Scalar Multiplication: Scalar multiplication by 1 must yield the original vector.

1 · a a for all a in V

1 can be defined as the multiplicative identity on the real number set, and it holds true that:

1 · (x1, y1) (x1, y1)

By verifying these postulates, we can conclude that the set V {x1y1 | xy∈R} satisfies all the necessary conditions to be considered a vector space under the defined operations of vector addition and scalar multiplication.

Conclusion

The set V {x1y1 | xy∈R} can indeed be made into a vector space if the appropriate definitions for vector addition and scalar multiplication are applied. Understanding the properties and axioms of vector spaces is crucial for various applications in mathematics and computer science, ranging from geometric transformations to machine learning algorithms.

By carefully defining the operations and verifying the 8 postulates, we have demonstrated the vector space properties of the set V. This knowledge can be further applied to other mathematical structures and operations, providing a solid foundation for advanced studies in linear algebra and related fields.