Problem 27 - Differential Equations and Linear Algebra

Question

Verify that the given set of objects together with the usual operations of addition and scalar multiplication is a complex vector space. \(M_{2}(\mathbb{C}),\) the set of all \(2 \times 2\) matrices with complex elements.

Step-by-Step Solution

Answer

\(M_{2}(\mathbb{C})\) is a complex vector space under the operations of addition and scalar multiplication since it satisfies all ten axioms of a complex vector space. These axioms include closure under addition and scalar multiplication, commutativity and associativity of addition, existence of additive identity and additive inverse, distributive properties with respect to scalar addition and matrix addition, scalar multiplication associativity, and scalar multiplication by the complex number 1.

Step 1: Axiom 1: Closure under addition

We need to show that if \(A\) and \(B\) are any two elements in \(M_{2}(\mathbb{C})\), then their sum, \(A+B\), also belongs to \(M_{2}(\mathbb{C})\). Let \(A\) and \(B\) be two arbitrary 2x2 matrices with complex elements: \[A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix},\] \[B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}.\] Now, considering the addition of these two matrices: \[A + B = \begin{pmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{pmatrix}.\] Since the sum of two complex numbers is still a complex number, each element in \(A+B\) is a complex number, and therefore \(A+B \in M_{2}(\mathbb{C})\). Axiom 1 is satisfied.

Step 2: Axiom 2: Commutativity of addition

We need to show that if \(A\) and \(B\) are any two elements in \(M_{2}(\mathbb{C})\), then \(A+B = B+A\). Using the definitions of \(A\) and \(B\) as given in Axiom 1: \[A + B = \begin{pmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{pmatrix} = \begin{pmatrix} b_{11} + a_{11} & b_{12} + a_{12} \\ b_{21} + a_{21} & b_{22} + a_{22} \end{pmatrix} = B + A.\] Axiom 2 is satisfied.

Step 3: Axiom 3: Associativity of addition

We need to show that if \(A\), \(B\), and \(C\) are any three elements in \(M_{2}(\mathbb{C})\), then \((A+B)+C = A+(B+C)\). Let \(C = \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix}\) be another 2x2 matrix with complex elements. Then, we can express the associative property: \[((A + B) + C) = \begin{pmatrix} (a_{11} + b_{11}) + c_{11} & (a_{12} + b_{12}) + c_{12} \\ (a_{21} + b_{21}) + c_{21} & (a_{22} + b_{22}) + c_{22} \end{pmatrix} = \begin{pmatrix} a_{11} + (b_{11} + c_{11}) & a_{12} + (b_{12} + c_{12}) \\ a_{21} + (b_{21} + c_{21}) & a_{22} + (b_{22} + c_{22}) \end{pmatrix} = (A + (B + C)).\] Axiom 3 is satisfied.

Step 4: Axiom 4: Existence of additive identity

We need to show that there exists a 2x2 matrix \(0\) with complex elements such that \(A+0=A\) for any \(A\) in \(M_{2}(\mathbb{C})\). Let \(0 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\). Then, for any \(A\) in \(M_{2}(\mathbb{C})\): \[A + 0 = \begin{pmatrix} a_{11} + 0 & a_{12} + 0 \\ a_{21} + 0 & a_{22} + 0 \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = A.\] Axiom 4 is satisfied.

Step 5: Axiom 5: Existence of additive inverse

We need to show that for each element \(A\) in \(M_{2}(\mathbb{C})\), there exists an element \(-A\) such that \(A + (-A) = 0\). Let \(-A = \begin{pmatrix} -a_{11} & -a_{12} \\ -a_{21} & -a_{22} \end{pmatrix}\). Then, for any \(A\) in \(M_{2}(\mathbb{C})\): \[A + (-A) = \begin{pmatrix} a_{11} + (-a_{11}) & a_{12} + (-a_{12}) \\ a_{21} + (-a_{21}) & a_{22} + (-a_{22}) \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = 0.\] Axiom 5 is satisfied.

Step 6: Axiom 6: Closure under scalar multiplication

We need to show that if \(A\) is an element in \(M_{2}(\mathbb{C})\) and \(c\) is a complex scalar, then the product \(cA\) also belongs to \(M_{2}(\mathbb{C})\). Let \(c \in \mathbb{C}\). Then: \[cA = c\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} ca_{11} & ca_{12} \\ ca_{21} & ca_{22} \end{pmatrix}.\] Since the product of a complex number and a complex scalar is still a complex number, each element in \(cA\) is a complex number, and therefore \(cA \in M_{2}(\mathbb{C})\). Axiom 6 is satisfied.

Step 7: Axiom 7: Distributive property with respect to scalar addition

We need to show that if \(A\) is an element in \(M_{2}(\mathbb{C})\) and \(c\) and \(d\) are complex scalars, then \((c+d)A = cA + dA\). Let \(c, d \in \mathbb{C}\). Then: \[((c+d)A) = (c+d)\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} (c+d)a_{11} & (c+d)a_{12} \\ (c+d)a_{21} & (c+d)a_{22} \end{pmatrix} = \begin{pmatrix} ca_{11} + da_{11} & ca_{12} + da_{12} \\ ca_{21} + da_{21} & ca_{22} + da_{22} \end{pmatrix} = c\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} + d\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = cA + dA.\] Axiom 7 is satisfied.

Step 8: Axiom 8: Distributive property with respect to matrix addition

We need to show that if \(A\) and \(B\) are elements in \(M_{2}(\mathbb{C})\) and \(c\) is a complex scalar, then \(c(A+B) = cA + cB\). Let \(c \in \mathbb{C}\). Then: \[c(A+B) = c\begin{pmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{pmatrix} = \begin{pmatrix} c(a_{11} + b_{11}) & c(a_{12} + b_{12}) \\ c(a_{21} + b_{21}) & c(a_{22} + b_{22}) \end{pmatrix} = \begin{pmatrix} ca_{11} + cb_{11} & ca_{12} + cb_{12} \\ ca_{21} + cb_{21} & ca_{22} + cb_{22} \end{pmatrix} = \begin{pmatrix} ca_{11} & ca_{12} \\ ca_{21} & ca_{22} \end{pmatrix} + \begin{pmatrix} cb_{11} & cb_{12} \\ cb_{21} & cb_{22} \end{pmatrix} = cA + cB.\] Axiom 8 is satisfied.

Step 9: Axiom 9: Scalar multiplication associates with scalar multiplication

We need to show that if \(A\) is an element in \(M_{2}(\mathbb{C})\) and \(c\) and \(d\) are complex scalars, then \((cd)A = c(dA)\). Let \(c, d \in \mathbb{C}\). Then: \[((cd)A) = (cd)\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} (cd)a_{11} & (cd)a_{12} \\ (cd)a_{21} & (cd)a_{22} \end{pmatrix} = c\begin{pmatrix} da_{11} & da_{12} \\ da_{21} & da_{22} \end{pmatrix} = c(dA).\] Axiom 9 is satisfied.

Step 10: Axiom 10: Scalar multiplication by the complex number 1

We need to show that if \(A\) is an element in \(M_{2}(\mathbb{C})\), then \(1A = A\), where 1 is the complex number 1 (the multiplicative identity). Then: \[1A = 1\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} 1a_{11} & 1a_{12} \\ 1a_{21} & 1a_{22} \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = A.\] Axiom 10 is satisfied. Since all ten axioms are satisfied, we can conclude that \(M_{2}(\mathbb{C})\) is a complex vector space under the operations of addition and scalar multiplication.

Key Concepts

Matrix Operations

In the context of a complex vector space like \(M_{2}(\mathbb{C})\), matrix operations play a crucial role. Matrix addition and scalar multiplication are essential operations that need to be understood well.

• **Addition of Matrices**: When adding two matrices, each corresponding element is added together. For instance, if \(A\) and \(B\) are both 2x2 matrices, then \(A + B\) results in another 2x2 matrix where each element is the sum of the corresponding elements of \(A\) and \(B\).
• **Scalar Multiplication**: This involves multiplying every element of a matrix by a scalar (a complex number, in this case). So, if \(c\) is a scalar and \(A\) is a matrix, \(cA\) means each element of \(A\) is multiplied by \(c\).

Understanding these operations helps verify the vector space axioms, as they are central to showing properties like closure and associativity. Matrices in complex vector spaces maintain their forms while these operations are performed, allowing for the axioms of vector spaces to be established.

Complex Numbers

Complex numbers extend the idea of one-dimensional number lines into two dimensions by including a real and an imaginary part, usually expressed in the form \(a + bi\) where \(a\) and \(b\) are real numbers and \(i\) is the square root of -1. This property of containing both real and imaginary parts allows complex numbers to be represented conveniently in 2D forms like matrices.

• **Properties of Complex Numbers**: Their addition and multiplication must adhere to specific rules due to their dual component structure. When adding, the real parts are added separately from the imaginary parts, maintaining the complex nature.
• **Applications in Matrices**: These numbers, when part of matrices, allow for enhanced manipulation, particularly useful in representing transformations in physics and engineering problems.

Complex numbers in matrices form the basis for operations in complex vector spaces, making it important to understand their foundational characteristics and how they interrelate with matrix components.

Vector Space Axioms

In algebra, vector spaces are defined by a set of rules or axioms which ensure structures like matrices behave consistently under operations like addition and scalar multiplication. For complex vector spaces, these axioms verify specific properties that guarantee the set's structure.

• **Closure Under Addition and Scalar Multiplication**: The set must be closed, meaning adding or multiplying elements within the set results in an element that is also in the set.
• **Existence of Additive Identity and Inverse**: There must be an additive identity (usually the zero matrix for matrices) so that any element plus this identity equals the element. Each element must also have an inverse, which negates it back to the identity.
• **Associativity and Distributivity**: These properties ensure that operations remain consistent regardless of how they are grouped or combined.

Understanding these axioms is key to grasping why \(M_{2}(\mathbb{C})\) behaves as a vector space under usual matrix operations. It is these properties that form the backbone of why matrices with complex elements act as valid vector space entities.