Problem 9 - 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 arrays of real numbers \(\left(\begin{array}{ll}a_{11} & a_{12} \\\ a_{21} & a_{22}\end{array}\right)\), addition and scalar multiplication defined by \(\begin{aligned}\left(\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right)+\left(\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right)=\left(\begin{array}{ll}a_{12}+b_{12} & a_{11}+b_{11} \\\ a_{22}+b_{22} & a_{21}+b_{21}\end{array}\right) \\ & k\left(\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right)=\left(\begin{array}{ll}k a_{11} & k a_{12} \\ k a_{21} & k a_{22}\end{array}\right) \end{aligned}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Operations

We are given a set of 2x2 arrays of real numbers with defined operations for addition and scalar multiplication. Addition is defined as \(\begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} + \begin{pmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{12} + b_{12} & a_{11} + b_{11} \ a_{22} + b_{22} & a_{21} + b_{21} \end{pmatrix}\). Scalar multiplication is the ordinary multiplication \( k\begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} ka_{11} & ka_{12} \ ka_{21} & ka_{22} \end{pmatrix} \).

Step 2: Check Closure Under Addition

We need to verify if the addition operation results in another 2x2 array of real numbers. For any arrays \(A\) and \(B\), the resulting array from addition is \(\begin{pmatrix} a_{12} + b_{12} & a_{11} + b_{11} \ a_{22} + b_{22} & a_{21} + b_{21} \end{pmatrix}\), which is still a 2x2 array of real numbers, satisfying the closure property for vectors.

Step 3: Check Commutative Property of Addition

For the set to be a vector space, addition must be commutative: \(A + B = B + A\). However, if we compute \(B + A = \begin{pmatrix} b_{12} + a_{12} & b_{11} + a_{11} \ b_{22} + a_{22} & b_{21} + a_{21} \end{pmatrix}\), this results in different arrays due to pairwise mismatching positions, failing the commutative property.

Step 4: Conclusion

Since the set of 2x2 arrays of real numbers with the given addition does not satisfy the commutative property of vector addition (\(A + B eq B + A\)), the set is not a vector space under the described operations.

Key Concepts

Closure Property

The closure property is a fundamental property of vector spaces, ensuring that when you perform vector addition or scalar multiplication on any vectors or elements in the space, the results are still within the space.

In the context of the provided exercise, closure under addition means that if we have two arrays, such as \[ \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \] plus \[ \begin{pmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{pmatrix}, \] the result must also be a 2x2 array of real numbers.

The exercise confirms this property: the sum \[ \begin{pmatrix} a_{12} + b_{12} & a_{11} + b_{11} \ a_{22} + b_{22} & a_{21} + b_{21} \end{pmatrix} \] is indeed still a 2x2 array, thus satisfying the closure property. This is crucial as, for a set to be a vector space, the results of all operations need to stay within the realm of the same set guide.

Commutative Property

The commutative property is a property of vector addition that states for any two vectors in a vector space, changing the order of vectors should not change the sum.

Mathematically, for vectors \( A \) and \( B \),
\( A + B = B + A \).

In the context of the exercise, checking commutativity means ensuring that whenever you add two 2x2 arrays as specified, the result is the same whichever array you start with. According to the provided solution, the operation defined in this exercise did not satisfy this property.

• Addition resulted in \[ \begin{pmatrix} a_{12} + b_{12} & a_{11} + b_{11} \ a_{22} + b_{22} & a_{21} + b_{21} \end{pmatrix}, \]
• Which is not equivalent to \[ \begin{pmatrix} b_{12} + a_{12} & b_{11} + a_{11} \ b_{22} + a_{22} & b_{21} + a_{21} \end{pmatrix}. \]

This failure breaks one of the core criteria for being a vector space and leads to the conclusion that the set fails to meet the requirements under the defined addition.

Vector Addition

Vector addition is the binary operation of adding two vectors to get another vector.

To be part of a vector space, addition needs to satisfy several properties, such as associativity and having an additive identity (zero vector).

In this exercise, vector addition is defined in an unconventional manner, resulting in the operations shifting element positions rather than maintaining them. Typically, you expect:

• Straightforward addition within corresponding parts: \( a_{11} + b_{11} \), \( a_{12} + b_{12} \)...
• Instead, the exercise uses a non-standard form, which rearranges components resulting in mismatched sums like \[ \begin{pmatrix} a_{12} + b_{12} & a_{11} + b_{11} \ a_{22} + b_{22} & a_{21} + b_{21} \end{pmatrix}. \]

Such rearrangements do not uphold the usual structure required for valid vector addition. Thus, in not maintaining standard form, it challenges the rules of vector space operations.

Scalar Multiplication

Scalar multiplication in vector spaces is an operation involving the multiplication of a vector by a scalar (a real number), resulting in another vector.

For a set to be a vector space, scalar multiplication must follow certain properties, such as distributivity and compatibility.

Within this exercise, scalar multiplication operates in a typical, expected manner:

• For a scalar \( k \) and an array \( \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \),
• The result is \( \begin{pmatrix} k \cdot a_{11} & k \cdot a_{12} \ k \cdot a_{21} & k \cdot a_{22} \end{pmatrix}. \)

This aligns with conventional vector space rules, demonstrating compatibility even though other operations did not conform. Hence, despite vector addition issues, scalar multiplication itself doesn't rule out the possibility of being part of a vector space.

Axioms of Vector Spaces

A vector space is defined by a series of axioms that guide its structure. These include properties about both vector addition and scalar multiplication.

Some important axioms include:

• Associativity of Addition: \( u + (v + w) = (u + v) + w \)
• Additive Identity: There exists a zero vector \( 0 \) such that \( u + 0 = u \)
• Additive Inverse: For every vector \( u \), there exists a \(-u \) such that \( u + (-u) = 0 \)
• Distributive Properties: \( a(u + v) = au + av \)and\((a + b)u = au + bu \)
• Compatibility with Scalar:\( ca(bu) = (ab)u \)

In assessing whether a set like the one in the exercise is a vector space, each axiom must be satisfied.
Due to the failure of the commutative property in the vector addition defined there, among others, the set does not meet vector space criteria. Each axiom provides fundamental rules, adherence to which is necessary for such classification.