Introduction to Vector Spaces of Complex Numbers
Understanding the concept of vector spaces is crucial for various fields, including mathematics, computer science, and engineering. In particular, the vector space formed by ordered pairs of complex numbers over the field of real numbers is a fascinating topic. This article explores the mathematical reasoning behind the dimension of such a vector space, providing clear explanations and rigorous proofs.
Representation of Complex Numbers in Real Coordinates
To begin, we need to express a complex number in terms of its real and imaginary components. A complex number (z) can be represented as:
[z a bi] where (a) and (b) are real numbers, and (i) is the imaginary unit, satisfying (i^2 -1).
Given an ordered pair of complex numbers (z_1) and (z_2), we can write:
[z_1 a_1 b_1i quad text{and} quad z_2 a_2 b_2i]These can be combined into a 4-tuple of real numbers:
[(a_1, b_1, a_2, b_2)]This representation shows that each ordered pair of complex numbers corresponds to a vector in (mathbb{R}^4), implying that the vector space of ordered pairs of complex numbers over the field of real numbers is isomorphic to (mathbb{R}^4).
Dimension of the Vector Space of Complex Pairs
The dimension of (mathbb{R}^4) as a vector space over the field of real numbers is 4. This is because (mathbb{R}^4) is a 4-dimensional vector space, meaning it has a basis consisting of 4 linearly independent vectors.
Therefore, the dimension of the vector space of ordered pairs of complex numbers over the field of real numbers is:
[boxed{4}]Operations in the Vector Space
When considering the complex numbers (mathbb{C}) as a vector space over the field of real numbers, there are two types of operations:
Addition of Complex Numbers as Ordered Pairs: ((a bi) (c di) (a c) (b d)i) Multiplication of Complex Numbers by Real Scalars: (t cdot (a bi) t(a bi) ta tbi)With these operations, (mathbb{C}) can be seen as a 2-dimensional vector space. The basis for this vector space is given by:
[B {1 0i, 0 1i} {1, i}]External Direct Sum and Dimension
Next, consider the vector space (V mathbb{C} times mathbb{C}). This vector space can be thought of as the external direct sum of two copies of (mathbb{C}). The operations in (V) are defined component-wise:
[t cdot (z_1, z_2) (t cdot z_1, t cdot z_2)]The external direct sum of vector spaces has a significant property: the dimension of the direct sum of two finite-dimensional vector spaces is the sum of their individual dimensions. Since (mathbb{C}) is a 2-dimensional vector space, the dimension of (V mathbb{C} oplus mathbb{C}) is:
[2 2 4]This confirms our earlier result that the dimension of the vector space of ordered pairs of complex numbers over the field of real numbers is (boxed{4}).