Is R2 A Subspace Of R3

The question of whether R² is a subspace of R³ is a fundamental concept in linear algebra. Understanding this requires a solid grasp of what subspaces are and how they relate to vector spaces. In essence, while R² and R³ are both vector spaces, R², as typically defined, isn't directly a subspace of R³. This stems from the fact that subspaces must be contained within the larger vector space, and elements of R² and R³ have different forms. However, R² can be represented as a subspace within R³. This distinction is crucial.

Defining Vector Spaces and Subspaces

To fully grasp the relationship between R² and R³, let's formally define vector spaces and subspaces.

Vector Spaces

A vector space is a set of objects (vectors) equipped with two operations: vector addition and scalar multiplication. These operations must satisfy a set of axioms to ensure that the structure behaves in a predictable way. These axioms ensure properties like closure under addition and scalar multiplication, existence of a zero vector, and existence of additive inverses.

Examples of vector spaces include:

• Rⁿ: The set of all n-tuples of real numbers, where addition and scalar multiplication are defined component-wise.
• The set of all polynomials with real coefficients.
• The set of all m x n matrices with real entries.

Subspaces

A subspace is a subset of a vector space that is itself a vector space under the same operations. For a subset W of a vector space V to be a subspace, it must satisfy three conditions:

1. The zero vector of V is in W.
2. If u and v are in W, then u + v is in W (closure under addition).
3. If u is in W and c is a scalar, then cu is in W (closure under scalar multiplication).

If a subset satisfies these three conditions, it forms a subspace. This means it "inherits" the vector space structure from its parent space.

Understanding R² and R³

Before we dive into whether R² is a subspace of R³, we need to clearly define these vector spaces.

R²: The Two-Dimensional Euclidean Space

R² represents the set of all ordered pairs of real numbers. In other words, it's the familiar Cartesian plane. Each element in R² is a vector of the form (x, y), where x and y are real numbers.

Vector addition in R² is defined as: (x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂)

Scalar multiplication is defined as: c(x, y) = (cx, cy)

R³: The Three-Dimensional Euclidean Space

R³ represents the set of all ordered triples of real numbers. This is the three-dimensional space we often visualize. Each element in R³ is a vector of the form (x, y, z), where x, y, and z are real numbers.

Vector addition in R³ is defined as: (x₁, y₁, z₁) + (x₂, y₂, z₂) = (x₁ + x₂, y₁ + y₂, z₁ + z₂)

Scalar multiplication is defined as: c(x, y, z) = (cx, cy, cz)

Why R² Isn't Directly a Subspace of R³

The most direct answer to the question is no, R² is not a subspace of R³. Here's why:

• Elements of R² and R³ have different forms. Elements in R² are ordered pairs (x, y), while elements in R³ are ordered triples (x, y, z). You can't directly say that (1, 2) is an element of R³ because it's missing the third component.
• Subspaces must be subsets. To be a subspace, all elements of the would-be subspace must also be elements of the larger vector space. Since elements of R² aren't directly elements of R³, R² cannot be a subspace of R³.

Representing R² as a Subspace of R³

While R² isn't inherently a subspace of R³, we can represent R² as a subspace within R³. This is done by embedding R² into R³.

The Embedding

We can define a mapping f: R² -> R³ such that: f(x, y) = (x, y, 0)

This mapping takes a vector in R² and maps it to a vector in R³ by adding a zero as the third component. The image of this mapping, which we'll call W, is the set of all vectors in R³ of the form (x, y, 0).

Proving W is a Subspace of R³

Now, let's show that W = {(x, y, 0) | x, y ∈ R} is a subspace of R³. We need to verify the three subspace conditions:

1. The zero vector of R³ is in W. The zero vector in R³ is (0, 0, 0). Since (0, 0, 0) can be written in the form (x, y, 0) where x = 0 and y = 0, it is in W.

2. If u and v are in W, then u + v is in W (closure under addition). Let u = (x₁, y₁, 0) and v = (x₂, y₂, 0) be two vectors in W. Then their sum is: u + v = (x₁ + x₂, y₁ + y₂, 0 + 0) = (x₁ + x₂, y₁ + y₂, 0) Since x₁ + x₂ and y₁ + y₂ are real numbers, u + v is also in the form (x, y, 0), and therefore u + v is in W.

3. If u is in W and c is a scalar, then cu is in W (closure under scalar multiplication). Let u = (x, y, 0) be a vector in W, and let c be a scalar. Then: cu = (cx, cy, c * 0) = (cx, cy, 0) Since cx and cy are real numbers, cu is also in the form (x, y, 0), and therefore cu is in W.

Since W satisfies all three conditions, W is a subspace of R³.

The Significance of the Embedding

This embedding is important because it allows us to treat R² as a "slice" of R³. Specifically, W can be visualized as the xy-plane within R³. Every point in the xy-plane has a z-coordinate of 0.

Isomorphism

The concept of an isomorphism further clarifies the relationship. An isomorphism is a bijective (one-to-one and onto) linear transformation between two vector spaces that preserves the vector space structure.

The mapping f: R² -> W defined as f(x, y) = (x, y, 0) is an isomorphism. This means that R² and W are essentially the same vector space, just represented differently. They have the same algebraic properties and structures.

To show that f is an isomorphism, we need to prove two things:

1. f is a linear transformation. This means that f(u + v) = f(u) + f(v) and f(cu) = cf(u) for all vectors u, v in R² and all scalars c.

   • Let u = (x₁, y₁) and v = (x₂, y₂). Then: f(u + v) = f(x₁ + x₂, y₁ + y₂) = (x₁ + x₂, y₁ + y₂, 0) f(u) + f(v) = (x₁, y₁, 0) + (x₂, y₂, 0) = (x₁ + x₂, y₁ + y₂, 0) Therefore, f(u + v) = f(u) + f(v).

   • Let u = (x, y) and c be a scalar. Then: f(cu) = f(cx, cy) = (cx, cy, 0) cf(u) = c(x, y, 0) = (cx, cy, 0) Therefore, f(cu) = cf(u).

   Since f satisfies both conditions, it is a linear transformation.

2. f is bijective (one-to-one and onto).

   • One-to-one (injective): If f(u) = f(v), then u = v. Suppose f(x₁, y₁) = f(x₂, y₂). Then (x₁, y₁, 0) = (x₂, y₂, 0). This implies that x₁ = x₂ and y₁ = y₂, so (x₁, y₁) = (x₂, y₂). Thus, f is one-to-one.

   • Onto (surjective): For every vector w in W, there exists a vector u in R² such that f(u) = w. Let w = (x, y, 0) be a vector in W. Then the vector u = (x, y) in R² satisfies f(u) = f(x, y) = (x, y, 0) = w. Thus, f is onto.

Since f is both a linear transformation and bijective, it is an isomorphism. This confirms that R² and W are isomorphic, meaning they are structurally equivalent.

Generalizing to Higher Dimensions

The same principle applies to higher dimensions. For example, Rⁿ can be represented as a subspace of R^m where m > n. We can define a mapping that takes a vector in Rⁿ and adds (m - n) zeros to create a vector in R^m. The image of this mapping will be a subspace of R^m isomorphic to Rⁿ.

Practical Applications

Understanding this concept has several practical applications:

• Computer Graphics: In computer graphics, 2D objects are often represented within a 3D space for manipulation and rendering. The 2D object is essentially treated as a subspace of the 3D space.
• Data Analysis: In data analysis, dimensionality reduction techniques often involve projecting high-dimensional data onto a lower-dimensional subspace. This allows for easier visualization and analysis while preserving the essential information.
• Physics: In physics, particularly in mechanics, constraints can reduce the degrees of freedom of a system. For instance, a particle moving on a plane (a 2D subspace) within 3D space is a common scenario.

Common Misconceptions

Several misconceptions often arise when discussing subspaces:

• Confusing subsets with subspaces: Not every subset of a vector space is a subspace. The subset must satisfy the three subspace conditions (containing the zero vector, closure under addition, and closure under scalar multiplication).
• Assuming all subspaces must pass through the origin: This is true because every subspace must contain the zero vector. If a subset doesn't contain the zero vector, it cannot be a subspace.
• Thinking isomorphism means equality: Isomorphism means that two vector spaces are structurally equivalent, not necessarily identical. They may have different representations but the same algebraic properties.

Examples

Let's solidify our understanding with some examples:

Example 1: The xy-plane in R³

As we've already discussed, the xy-plane in R³, defined as W = {(x, y, 0) | x, y ∈ R}, is a subspace of R³. This is a common and intuitive example.

Example 2: A line through the origin in R²

Consider the line L defined by y = 2x in R². We can write this as L = {(x, 2x) | x ∈ R}. Let's check if L is a subspace of R²:

1. The zero vector (0, 0) is in L. Since 0 = 2 * 0, (0, 0) is in L.
2. Closure under addition: Let u = (x₁, 2x₁) and v = (x₂, 2x₂) be in L. Then u + v = (x₁ + x₂, 2x₁ + 2x₂) = (x₁ + x₂, 2(x₁ + x₂)). Since the second component is twice the first component, u + v is in L.
3. Closure under scalar multiplication: Let u = (x, 2x) be in L and c be a scalar. Then cu = (cx, 2cx). Again, the second component is twice the first component, so cu is in L.

Therefore, L is a subspace of R².

Example 3: A plane in R³ passing through the origin

Consider the plane P in R³ defined by the equation x + y + z = 0. We can write this as P = {(x, y, z) | x + y + z = 0}. Let's check if P is a subspace of R³:

1. The zero vector (0, 0, 0) is in P. Since 0 + 0 + 0 = 0, (0, 0, 0) is in P.
2. Closure under addition: Let u = (x₁, y₁, z₁) and v = (x₂, y₂, z₂) be in P. Then x₁ + y₁ + z₁ = 0 and x₂ + y₂ + z₂ = 0. Then u + v = (x₁ + x₂, y₁ + y₂, z₁ + z₂). Adding the equations, we get (x₁ + x₂) + (y₁ + y₂) + (z₁ + z₂) = 0 + 0 = 0. Therefore, u + v is in P.
3. Closure under scalar multiplication: Let u = (x, y, z) be in P and c be a scalar. Then x + y + z = 0. Then cu = (cx, cy, cz). Multiplying the equation by c, we get cx + cy + cz = c(x + y + z) = c * 0 = 0. Therefore, cu is in P.

Therefore, P is a subspace of R³.

Conclusion

While R² is not directly a subspace of R³ because its elements have a different form, it can be represented as a subspace of R³ through an embedding. This embedded subspace, often visualized as the xy-plane, is isomorphic to R², meaning they are structurally equivalent. Understanding this distinction requires a solid grasp of vector spaces, subspaces, and the concept of isomorphism. This knowledge is fundamental in linear algebra and has practical applications in various fields like computer graphics, data analysis, and physics. Remembering the key conditions for a subset to be a subspace—containing the zero vector, closure under addition, and closure under scalar multiplication—is crucial for correctly identifying subspaces within larger vector spaces.