Final answer:
A complex number, expressed as a + bi, can be considered as a 2D vector space over the scalar field of real numbers. The vector addition and scalar multiplication operations of complex numbers satisfy the necessary properties for a vector space.
Step-by-step explanation:
In this case, we consider the field of complex numbers, denoted by ℝ. A complex number consists of a real part and an imaginary part, where the imaginary part is a multiple of the imaginary unit i. For example, a complex number can be expressed as a + bi, where a and b are real numbers. The set of complex numbers forms a vector space over the field of real numbers, which means it satisfies the necessary properties for a vector space.
To show that ℝ is a 2D vector space over the scalar field ℝ, we need to show that it satisfies the vector space axioms. The vector addition and scalar multiplication operations of complex numbers satisfy the commutative, associative, distributive, and zero vector properties, which are necessary for a vector space. Therefore, we can conclude that ℝ is a 2D vector space over ℝ with the standard addition and multiplication of complex numbers.