Problem 7 - Advanced Engineering Mathematics

Question

In Problems 1-10, determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless stated to the contrary, assume that vector addition and scalar multiplication are the ordinary operations defined on that set. The set of real numbers, addition defined by \(\mathbf{x}+\mathbf{y}=x-y\)

Step-by-Step Solution

Answer

The set is not a vector space because addition is not commutative or associative.

Step 1: List Vector Space Axioms

A set is considered a vector space if it satisfies the following axioms with respect to addition and scalar multiplication: closure under addition and scalar multiplication, associativity of addition, commutativity of addition, existence of additive identity, existence of additive inverses, distributivity of scalar multiplication with respect to vector addition, distributivity of scalar multiplication with respect to field addition, compatibility of scalar multiplication, and existence of multiplicative identity for scalar multiplication.

Step 2: Test Closure Under Addition

For closure under addition, take two real numbers \(a\) and \(b\). When we define addition as \(a + b = a - b\), the result is not necessarily a real number since subtraction is not the same as usual addition. Although real numbers are closed under subtraction, we need to check if this operation satisfies all other axioms.

Step 3: Test Commutativity of Addition

For commutativity, check if \(a + b = b + a\). With our operation \(a + b = a - b\), we have \(a - b\) and \(b - a\). Since these are not always equal (e.g., \(3 - 5 eq 5 - 3\)), the operation is not commutative.

Step 4: Test Associativity of Addition

For associativity, check if \((a + b) + c = a + (b + c)\). Using the defined operation, \((a + b) + c = (a - b) - c\) and \(a + (b + c) = a - (b - c)\). These simplify to \(a - b - c\) and \(a - b + c\) respectively, which are clearly not equal in general. Thus, addition is not associative.

Step 5: Identify Non-Satisfied Axiom

Based on Steps 3 and 4, we've identified that the defined operation fails the commutativity and associativity axioms of vector addition. Since not all axioms are satisfied, this set cannot be a vector space.

Key Concepts

Vector Addition

In the context of vector spaces, vector addition is a fundamental operation that combines two vectors to produce a third vector. Usually, vector addition should satisfy three main properties:

• Commutativity: This property means that the order in which vectors are added does not matter. Mathematically, for any two vectors, \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \).
• Associativity: This suggests that when three vectors are added, it doesn't matter how they are grouped. So, \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \).
• Additive Identity: There must exist a vector (typically the zero vector) such that when added to any vector, it returns the original vector. For vector \( \mathbf{v} \), \( \mathbf{v} + \mathbf{0} = \mathbf{v} \).

In the provided problem, vector addition was redefined as subtraction \( a + b = a - b \). This operation fails some core properties of vector addition. For instance, it is not commutative, because subtracting two numbers in different orders gives different results. Similarly, it is not associative, as grouping terms will lead to different outcomes. These failures disqualify the set from being a vector space.

Scalar Multiplication

Scalar multiplication in a vector space combines a scalar (a real number) with a vector to produce another vector. This operation should follow specific rules to maintain the structure of a vector space:

• Distributive Property 1: Scalar multiplication should distribute over vector addition, meaning \( a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v} \).
• Distributive Property 2: It should also distribute over field addition, ensuring \( (a + b)\mathbf{v} = a\mathbf{v} + b\mathbf{v} \).
• Compatibility: The operation must align with field multiplication such that \( a(b\mathbf{v}) = (ab)\mathbf{v} \).
• Multiplicative Identity: There must be a scalar multiplicative identity, typically 1, where \( 1\cdot\mathbf{v} = \mathbf{v} \).

Each of these properties ensures that scalar multiplication interacts smoothly with vector addition, maintaining the vector's properties. In our problem's specific context, scalar multiplication with subtraction defined as addition might still hold true for some parts but wouldn't establish a vector space without proper vector addition.

Vector Space Axioms

To formally identify a vector space, a set must satisfy specific axioms related to vector addition and scalar multiplication. Here's a summary:

• Closure under Addition and Scalar Multiplication: The operation results should always remain within the set.
• Associativity and Commutativity of Addition: The usual properties we expect with addition, ensuring consistent results.
• Existence of Additive Identity and Inverses: The zero vector must exist, and each vector should have an inverse that nullifies it when added together.
• Distributive Properties: Both vectors and scalars should distribute properly, aligning their respective operations.
• Existence of Multiplicative Identity: Similar to the additive identity, ensuring scalar multiplication keeps the vector intact when multiplied by one.

In our problem, the unconventional definition of vector addition disrupts several key axioms, specifically those concerning addition's commutativity and associativity. Failing these axioms precludes the set from being defined as a vector space. Understanding and identifying which axiom fails is crucial when determining vector spaces.