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Chapter 6: Problem 25

Determine whether \(\mathrm{V}\) and \(\mathrm{W}\) are isomorphic. If they are, give an explicit isomorphism \(T: V \rightarrow W\) \(V=\mathbb{C}, W=\mathbb{R}^{2}\)

Short Answer

Expert verified
Yes, \\(V\\) and \\(W\\) are isomorphic; an isomorphism is given by \\(T(a + bi) = (a, b)\\).

Step by step solution

01

Define the Spaces

We need to understand the structure and properties of the spaces involved. \(V\) is defined as \(\mathbb{C}\), the set of complex numbers, which can be viewed as a 2-dimensional vector space over \(\mathbb{R}\). \(W\) is \(\mathbb{R}^{2}\), a 2-dimensional real vector space.
02

Determine Dimensions

For vector spaces to be isomorphic, they must have the same dimension. We know both \(\mathbb{C}\) (as a 2D real vector space) and \(\mathbb{R}^2\) have dimension 2 over \(\mathbb{R}\). This condition for isomorphism is satisfied.
03

Construct an Isomorphism

To explicitly construct an isomorphism \(T: V \to W\), consider any complex number \(z = a + bi\) where \(a, b \in \mathbb{R}\). We can map \(z\) to a real 2D vector \((a, b)\) in \(\mathbb{R}^{2}\).
04

Verify Isomorphism Property

Check if the mapping \(T(z) = (a, b)\) satisfies the properties of an isomorphism: linearity and bijectiveness. Linearity: \(T(cz_1 + z_2) = cT(z_1) + T(z_2)\). Bijectiveness: Clearly, the mapping is both injective (no two different complex numbers map to the same vector) and surjective (every vector in \(\mathbb{R}^2\) comes from some complex number), confirming the isomorphism.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are an extended set of numbers that include all real numbers, plus an additional unit called the imaginary unit, denoted as "i." The basic form of a complex number is \( z = a + bi \), where \( a \) and \( b \) are real numbers. Here:
  • \( a \) is the real part of the complex number.
  • \( bi \) is the imaginary part, where \( i \) is the imaginary unit defined by \( i^2 = -1 \).
Imaginary numbers allow us to solve equations that have no real solution, such as \( x^2 + 1 = 0 \). Complex numbers can be visualized on a plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This plane is known as the complex plane or Argand plane. They play a critical role in solving problems in engineering, physics, and computer science.
Real Vector Spaces
A real vector space is a collection of objects called vectors, where addition and scalar multiplication are defined and follow certain rules. These spaces use real numbers as the field of scalars. Real vector spaces can be visualized in \( n \)-dimensional real space, denoted \( \mathbb{R}^n \).Importantly, vector spaces must satisfy the following axioms:
  • Closure under addition and scalar multiplication.
  • Existence of a zero vector that exits for any vector in the space.
  • Associative and commutative properties of vector addition.
  • Distributive, associative and multiplicative identity properties of scalar multiplication.
For example, \( \mathbb{R}^2 \) is a 2-dimensional real vector space, containing all pairs of real numbers \((x, y)\). Vectors can be added together or multiplied by real numbers (scalars), producing another vector within the same space.
Dimension
The dimension of a vector space is a measure of its "size" or "complexity." It is defined as the maximum number of linearly independent vectors in the space. Familiar spaces like \( \mathbb{R}^n \) have dimension \( n \). For instance:
  • \( \mathbb{R}^2 \) has a dimension of 2, with vectors \( (1, 0) \) and \( (0, 1) \) forming a basis.
  • The complex numbers \( \mathbb{C} \), when viewed as a real vector space, have dimension 2, as each complex number \( a + bi \) can be represented by the real vectors \( (a, b) \).
In isomorphic vector spaces, dimension plays a crucial role, as a necessary condition for two spaces to be isomorphic is that they must have the same dimension over the same field. In our exercise, both \( \mathbb{C} \) as a real vector space and \( \mathbb{R}^2 \) have dimension 2 over \( \mathbb{R} \), fulfilling this condition.
Linear Maps
Linear maps, or linear transformations, are functions that preserve the operations of vector addition and scalar multiplication. A map \( T: V \rightarrow W \) between vector spaces \( V \) and \( W \) is linear if:
  • \( T(u + v) = T(u) + T(v) \) for all vectors \( u, v \) in \( V \).
  • \( T(cv) = cT(v) \) for any scalar \( c \) and vector \( v \) in \( V \).
For an isomorphism, which is a special type of linear map, the map must also be bijective, meaning:
  • Injective: Every element in \( V \) maps to a unique element in \( W \).
  • Surjective: Every element in \( W \) is an image of some element in \( V \).
In the solution exercise, the map \( T: \mathbb{C} \rightarrow \mathbb{R}^2 \) defined as \( T(a + bi) = (a, b) \) is demonstrated to be a linear isomorphism, since it satisfies both linearity and bijectiveness. This confirms that \( \mathbb{C} \) and \( \mathbb{R}^2 \) are isomorphic as real vector spaces.

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