Which Of The Following Sets Are Subspaces Of R3

Determining Subspaces of R³: A Comprehensive Guide

Understanding subspaces is crucial in linear algebra, forming the foundation for many advanced concepts. This article will guide you through the process of identifying whether a given set is a subspace of R³, the set of all three-dimensional real vectors. We'll explore the criteria for subspace qualification, work through several examples, and address common misconceptions. By the end, you’ll be equipped to confidently determine whether any given set forms a subspace of R³.

Introduction to Subspaces of R³

A subspace of R³ is a subset of R³ that satisfies three specific conditions: it must contain the zero vector, be closed under addition, and be closed under scalar multiplication. Let's break down each of these conditions:

1. Contains the Zero Vector: The zero vector, denoted as 0 = (0, 0, 0), must be an element of the subset. This is a fundamental requirement.

2. Closure under Addition: If vectors u and v are elements of the subset, then their sum, u + v, must also be an element of the subset. In other words, adding any two vectors within the subset keeps you within the subset.

3. Closure under Scalar Multiplication: If vector u is an element of the subset, and c is any scalar (a real number), then the scalar multiple cu must also be an element of the subset. This means multiplying any vector in the subset by a scalar keeps you within the subset.

Methods for Determining Subspaces

To determine if a given set is a subspace of R³, we need to systematically check each of the three conditions above. Let's illustrate this with some examples.

Example 1: The Set of All Vectors of the Form (x, 0, 0)

Let's consider the set S₁ = {(x, 0, 0) | x ∈ ℝ}. This set represents all vectors in R³ where the y and z components are zero.

1. Zero Vector: The zero vector (0, 0, 0) is in S₁ (when x = 0).

2. Closure under Addition: Let u = (x₁, 0, 0) and v = (x₂, 0, 0) be two vectors in S₁. Their sum is u + v = (x₁ + x₂, 0, 0). Since x₁ + x₂ is a real number, this sum is also in S₁.

3. Closure under Scalar Multiplication: Let u = (x, 0, 0) be a vector in S₁ and c be a scalar. Then cu = (cx, 0, 0). Again, cx is a real number, so cu is in S₁.

Conclusion: Since S₁ satisfies all three conditions, it is a subspace of R³. Geometrically, S₁ represents the x-axis in three-dimensional space.

Example 2: The Set of All Vectors of the Form (1, y, z)

Let's examine the set S₂ = {(1, y, z) | y, z ∈ ℝ}. This set comprises all vectors with a fixed x-component of 1.

1. Zero Vector: The zero vector (0, 0, 0) is not in S₂. There is no combination of y and z that will make the x-component zero.

Conclusion: Since S₂ fails the first condition, it is not a subspace of R³.

Example 3: The Set of All Vectors (x, y, z) such that x + y + z = 1

Consider the set S₃ = {(x, y, z) | x + y + z = 1, x, y, z ∈ ℝ}. This set contains all vectors whose components sum to 1.

1. Zero Vector: The zero vector (0, 0, 0) is not in S₃ because 0 + 0 + 0 ≠ 1.

Conclusion: Because S₃ fails the first condition, it is not a subspace of R³.

Example 4: The Set of All Vectors (x, y, z) such that x + y + z = 0

Let's analyze S₄ = {(x, y, z) | x + y + z = 0, x, y, z ∈ ℝ}. This set includes all vectors whose components sum to zero.

1. Zero Vector: The zero vector (0, 0, 0) is in S₄ because 0 + 0 + 0 = 0.

2. Closure under Addition: Let u = (x₁, y₁, z₁) and v = (x₂, y₂, z₂) be in S₄. This means x₁ + y₁ + z₁ = 0 and x₂ + y₂ + z₂ = 0. Their sum is u + v = (x₁ + x₂, y₁ + y₂, z₁ + z₂). Adding the components, we get (x₁ + x₂) + (y₁ + y₂) + (z₁ + z₂) = (x₁ + y₁ + z₁) + (x₂ + y₂ + z₂) = 0 + 0 = 0. Thus, u + v is in S₄.

3. Closure under Scalar Multiplication: Let u = (x, y, z) be in S₄ (so x + y + z = 0), and let c be a scalar. Then cu = (cx, cy, cz). Adding the components, we have cx + cy + cz = c(x + y + z) = c(0) = 0. Therefore, cu is in S₄.

Conclusion: S₄ satisfies all three conditions and is a subspace of R³. Geometrically, this represents a plane passing through the origin.

Example 5: The Set of All Vectors (x, y, z) such that x² + y² + z² = 1

Let's consider S₅ = {(x, y, z) | x² + y² + z² = 1, x, y, z ∈ ℝ}. This set represents all vectors with a magnitude (length) of 1—the unit sphere centered at the origin.

1. Zero Vector: The zero vector (0, 0, 0) is not in S₅ because 0² + 0² + 0² ≠ 1.

Conclusion: S₅ is not a subspace of R³ because it does not contain the zero vector.

Example 6: The Set of All Linear Combinations of Two Linearly Independent Vectors

Let's consider two linearly independent vectors in R³, u = (1, 0, 0) and v = (0, 1, 0). The set S₆ is defined as all linear combinations of u and v: S₆ = {au + bv | a, b ∈ ℝ}.

1. Zero Vector: When a = 0 and b = 0, we get the zero vector (0, 0, 0).

2. Closure under Addition: Let w₁ = a₁u + b₁v and w₂ = a₂u + b₂v be two vectors in S₆. Their sum is w₁ + w₂ = (a₁ + a₂)u + (b₁ + b₂)v, which is also a linear combination of u and v, and thus in S₆.

3. Closure under Scalar Multiplication: Let w = au + bv be a vector in S₆, and let c be a scalar. Then cw = (ca)u + (cb)v, which is again a linear combination of u and v, and therefore in S₆.

Conclusion: S₆ is a subspace of R³. Geometrically, this represents the xy-plane.

Frequently Asked Questions (FAQ)

Q1: What does "linearly independent" mean in the context of subspaces?

A1: Two vectors are linearly independent if neither is a scalar multiple of the other. In simpler terms, they point in different directions and cannot be expressed as a multiple of each other. This is a crucial concept when defining subspaces spanned by a set of vectors.

Q2: Can a subspace contain only the zero vector?

A2: Yes, the set containing only the zero vector, {0}, is a subspace of any vector space, including R³. It trivially satisfies all three conditions.

Q3: Is the empty set a subspace?

A3: No, the empty set is not considered a subspace. Subspaces must contain the zero vector.

Q4: How can I visualize subspaces of R³?

A4: Subspaces of R³ can be visualized geometrically. The simplest subspaces are points (the zero vector), lines passing through the origin, and planes passing through the origin. More complex subspaces can be thought of as combinations of these.

Conclusion

Determining whether a set is a subspace of R³ involves a systematic check of three conditions: it must contain the zero vector, be closed under addition, and be closed under scalar multiplication. By carefully applying these conditions, you can confidently identify subspaces of R³ and gain a deeper understanding of the fundamental structures of linear algebra. Remember to analyze each set individually, carefully considering whether it satisfies all three criteria. This methodical approach will ensure accurate determination of subspaces and enhance your grasp of this crucial concept. Mastering this skill is essential for further exploration into linear algebra's vast and powerful applications.