What Is Subspace In Linear Algebra

In linear algebra, a subspace is a subset of a vector space that itself satisfies the axioms of a vector space. This concept is fundamental to understanding the structure and properties of vector spaces, providing a framework for analyzing linear transformations and solving systems of linear equations. Understanding subspaces is crucial for various applications, including computer graphics, data analysis, and engineering.

Defining Subspace

A subspace, denoted as W, of a vector space V over a field F (e.g., real numbers ℝ or complex numbers ℂ), must satisfy the following three conditions:

1. Non-empty: W must contain the zero vector of V. This ensures that W is not an empty set and has a starting point.
2. Closure under addition: For any two vectors u and v in W, their sum (u + v) must also be in W. This means that W is "closed" under the addition operation defined in V.
3. Closure under scalar multiplication: For any vector u in W and any scalar c in F, the scalar product (cu) must also be in W. This means that W is "closed" under scalar multiplication defined in V.

If a subset W of a vector space V satisfies these three conditions, then W is a subspace of V. This implies that W is itself a vector space, inheriting the operations of vector addition and scalar multiplication from V.

Formal Definition

Let V be a vector space over a field F. A subset W of V is a subspace of V if and only if:

1. 0 ∈ W (Zero vector condition)
2. For all u, v ∈ W, u + v ∈ W (Closure under addition)
3. For all u ∈ W and c ∈ F, cu ∈ W (Closure under scalar multiplication)

Examples of Subspaces

To illustrate the concept of subspaces, let's explore several examples within different vector spaces.

1. The Zero Subspace

In any vector space V, the set containing only the zero vector, {0}, is always a subspace. This is known as the zero subspace.

• Zero vector condition: The zero vector is in the set.
• Closure under addition: 0 + 0 = 0, which is in the set.
• Closure under scalar multiplication: c0 = 0 for any scalar c, which is in the set.

2. The Vector Space Itself

The entire vector space V is always a subspace of itself.

• Zero vector condition: V contains the zero vector.
• Closure under addition: For any u, v ∈ V, u + v ∈ V.
• Closure under scalar multiplication: For any u ∈ V and c, cu ∈ V.

3. Subspaces of ℝ²

Consider the vector space ℝ², which represents the set of all two-dimensional vectors with real number components.

• A line through the origin: Any line passing through the origin (0, 0) is a subspace of ℝ². For example, the line y = 2x.
  • The point (0, 0) lies on the line.
  • If (u₁, u₂) and (v₁, v₂) are points on the line, then u₂ = 2u₁ and v₂ = 2v₁. Their sum is (u₁ + v₁, u₂ + v₂) = (u₁ + v₁, 2(u₁ + v₁)), which also lies on the line.
  • If (u₁, u₂) is a point on the line, then u₂ = 2u₁. For any scalar c, cu = (cu₁, cu₂) = (cu₁, 2(cu₁)), which also lies on the line.
• A line not through the origin: A line that does not pass through the origin is not a subspace of ℝ². For example, the line y = 2x + 1.
  • The point (0, 0) does not lie on this line, violating the zero vector condition.

4. Subspaces of ℝ³

Consider the vector space ℝ³, which represents the set of all three-dimensional vectors with real number components.

• A plane through the origin: Any plane passing through the origin (0, 0, 0) is a subspace of ℝ³. For example, the plane x + y - z = 0.
  • The point (0, 0, 0) lies on the plane.
  • If (u₁, u₂, u₃) and (v₁, v₂, v₃) are points on the plane, then u₁ + u₂ - u₃ = 0 and v₁ + v₂ - v₃ = 0. Their sum is (u₁ + v₁, u₂ + v₂, u₃ + v₃), and (u₁ + v₁) + (u₂ + v₂) - (u₃ + v₃) = (u₁ + u₂ - u₃) + (v₁ + v₂ - v₃) = 0 + 0 = 0, so the sum lies on the plane.
  • If (u₁, u₂, u₃) is a point on the plane, then u₁ + u₂ - u₃ = 0. For any scalar c, cu = (cu₁, cu₂, cu₃), and cu₁ + cu₂ - cu₃ = c(u₁ + u₂ - u₃) = c(0) = 0, so the scalar product lies on the plane.
• A plane not through the origin: A plane that does not pass through the origin is not a subspace of ℝ³. For example, the plane x + y - z = 1.
  • The point (0, 0, 0) does not lie on this plane, violating the zero vector condition.
• A line through the origin: Any line passing through the origin (0, 0, 0) is a subspace of ℝ³. For example, the line defined by x = t, y = 2t, z = 3t, where t is a parameter.
  • When t = 0, we get the point (0, 0, 0), which lies on the line.
  • If u = (t₁, 2t₁, 3t₁) and v = (t₂, 2t₂, 3t₂) are points on the line, their sum is u + v = (t₁ + t₂, 2(t₁ + t₂), 3(t₁ + t₂)), which also lies on the line.
  • If u = (t, 2t, 3t) is a point on the line, for any scalar c, cu = (ct, 2ct, 3c*t), which also lies on the line.

5. Subspaces of Polynomials

Consider the vector space Pₙ of all polynomials of degree at most n with real coefficients.

• The set of all polynomials of degree at most m, where m < n, is a subspace of Pₙ. For example, the set of all quadratic polynomials is a subspace of the set of all cubic polynomials.
• The set of all polynomials with a zero constant term is a subspace.
  • The zero polynomial has a zero constant term.
  • The sum of two polynomials with zero constant terms also has a zero constant term.
  • A scalar multiple of a polynomial with a zero constant term also has a zero constant term.
• The set of all polynomials with a non-zero constant term is not a subspace, as it does not contain the zero polynomial.

How to Prove a Subset is a Subspace

To rigorously prove that a subset W of a vector space V is a subspace, you must verify all three conditions:

1. Verify the zero vector condition: Show that the zero vector 0 of V is an element of W.
2. Verify closure under addition: Take any two arbitrary vectors u and v in W. Show that their sum (u + v) is also in W. This often involves using the properties that define W to demonstrate that the sum satisfies the same properties.
3. Verify closure under scalar multiplication: Take any arbitrary vector u in W and any scalar c in the field F. Show that the scalar product (cu) is also in W. This often involves using the properties that define W to demonstrate that the scalar product satisfies the same properties.

If all three conditions hold, then W is a subspace of V. If any one of these conditions fails, then W is not a subspace of V.

Span and Subspaces

The span of a set of vectors in a vector space V is the set of all possible linear combinations of those vectors. More formally, if S = {v₁, v₂, ..., vₙ} is a set of vectors in V, then the span of S, denoted as span(S), is defined as:

span(S) = {c₁v₁ + c₂v₂ + ... + cₙvₙ | c₁, c₂, ..., cₙ ∈ F}

A crucial result is that the span of any set of vectors in a vector space is always a subspace. This provides a powerful way to generate subspaces.

Theorem: The Span is a Subspace

If S = {v₁, v₂, ..., vₙ} is a set of vectors in a vector space V, then span(S) is a subspace of V.

Proof:

1. Zero vector condition: The zero vector is in span(S), since 0v₁ + 0v₂ + ... + 0vₙ = 0.
2. Closure under addition: Let u and w be in span(S). Then, u = a₁v₁ + a₂v₂ + ... + aₙvₙ for some scalars a₁, a₂, ..., aₙ in F, and w = b₁v₁ + b₂v₂ + ... + bₙvₙ for some scalars b₁, b₂, ..., bₙ in F. Then, u + w = (a₁ + b₁) v₁ + (a₂ + b₂) v₂ + ... + (aₙ + bₙ) vₙ, which is a linear combination of the vectors in S, and thus u + w is in span(S).
3. Closure under scalar multiplication: Let u be in span(S) and let c be a scalar in F. Then, u = a₁v₁ + a₂v₂ + ... + aₙvₙ for some scalars a₁, a₂, ..., aₙ in F. Then, cu = (ca₁) v₁ + (ca₂) v₂ + ... + (caₙ) vₙ, which is a linear combination of the vectors in S, and thus cu* is in span(S).

Therefore, span(S) is a subspace of V.

Example of Span

In ℝ³, consider the vectors v₁ = (1, 0, 0) and v₂ = (0, 1, 0). The span of these two vectors, span({v₁, v₂}), is the set of all linear combinations of v₁ and v₂:

span({v₁, v₂}) = {a(1, 0, 0) + b(0, 1, 0) | a, b ∈ ℝ} = {(a, b, 0) | a, b ∈ ℝ}

This is the xy-plane, which is a subspace of ℝ³.

Linear Independence and Basis

Understanding linear independence and basis is crucial for grasping the structure of subspaces.

Linear Independence

A set of vectors {v₁, v₂, ..., vₙ} in a vector space V is said to be linearly independent if the only solution to the equation

c₁v₁ + c₂v₂ + ... + cₙvₙ = 0

is c₁ = c₂ = ... = cₙ = 0. In other words, no vector in the set can be written as a linear combination of the other vectors.

If the set of vectors is not linearly independent, it is said to be linearly dependent. This means that there exist scalars c₁, c₂, ..., cₙ, not all zero, such that

c₁v₁ + c₂v₂ + ... + cₙvₙ = 0

Basis

A basis of a vector space V is a set of vectors that is both linearly independent and spans V. A basis provides a minimal set of vectors that can be used to generate the entire vector space through linear combinations.

Properties of a Basis

• Every vector in V can be written as a unique linear combination of the basis vectors.
• All bases of a given vector space have the same number of vectors. This number is called the dimension of the vector space.

Example of a Basis

In ℝ³, the standard basis is {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. This set of vectors is linearly independent and spans ℝ³. Therefore, it is a basis for ℝ³, and the dimension of ℝ³ is 3.

Basis of a Subspace

A subspace W of a vector space V is itself a vector space, so it also has a basis. The basis of a subspace is a set of linearly independent vectors that span the subspace.

Example

Consider the subspace W of ℝ³ defined by the equation x + y + z = 0. A basis for this subspace is { (1, -1, 0), (1, 0, -1) }. These two vectors are linearly independent and any vector in W can be written as a linear combination of these two vectors. The dimension of W is 2.

Intersection and Sum of Subspaces

Subspaces can be combined in various ways to form new subspaces. Two common operations are the intersection and sum of subspaces.

Intersection of Subspaces

If W₁ and W₂ are subspaces of a vector space V, then their intersection, denoted as W₁ ∩ W₂, is the set of all vectors that are in both W₁ and W₂.

Theorem: The Intersection is a Subspace

If W₁ and W₂ are subspaces of a vector space V, then W₁ ∩ W₂ is also a subspace of V.

Proof:

1. Zero vector condition: Since W₁ and W₂ are subspaces, 0 ∈ W₁ and 0 ∈ W₂. Therefore, 0 ∈ W₁ ∩ W₂.
2. Closure under addition: Let u, v ∈ W₁ ∩ W₂. Then u, v ∈ W₁ and u, v ∈ W₂. Since W₁ and W₂ are subspaces, u + v ∈ W₁ and u + v ∈ W₂. Therefore, u + v ∈ W₁ ∩ W₂.
3. Closure under scalar multiplication: Let u ∈ W₁ ∩ W₂ and let c be a scalar in F. Then u ∈ W₁ and u ∈ W₂. Since W₁ and W₂ are subspaces, cu ∈ W₁ and cu ∈ W₂. Therefore, cu ∈ W₁ ∩ W₂.

Therefore, W₁ ∩ W₂ is a subspace of V.

Sum of Subspaces

If W₁ and W₂ are subspaces of a vector space V, then their sum, denoted as W₁ + W₂, is the set of all vectors that can be written as the sum of a vector in W₁ and a vector in W₂.

W₁ + W₂ = {u + v | u ∈ W₁, v ∈ W₂}

Theorem: The Sum is a Subspace

If W₁ and W₂ are subspaces of a vector space V, then W₁ + W₂ is also a subspace of V.

Proof:

1. Zero vector condition: Since W₁ and W₂ are subspaces, 0 ∈ W₁ and 0 ∈ W₂. Therefore, 0 + 0 = 0 ∈ W₁ + W₂.
2. Closure under addition: Let x, y ∈ W₁ + W₂. Then x = u₁ + v₁ for some u₁ ∈ W₁ and v₁ ∈ W₂, and y = u₂ + v₂ for some u₂ ∈ W₁ and v₂ ∈ W₂. Then x + y = (u₁ + u₂) + (v₁ + v₂). Since W₁ and W₂ are subspaces, u₁ + u₂ ∈ W₁ and v₁ + v₂ ∈ W₂. Therefore, x + y ∈ W₁ + W₂.
3. Closure under scalar multiplication: Let x ∈ W₁ + W₂ and let c be a scalar in F. Then x = u + v for some u ∈ W₁ and v ∈ W₂. Then cx = c(u + v) = cu + cv. Since W₁ and W₂ are subspaces, cu ∈ W₁ and cv ∈ W₂. Therefore, cx ∈ W₁ + W₂.

Therefore, W₁ + W₂ is a subspace of V.

Direct Sum

The sum W₁ + W₂ is called a direct sum, denoted as W₁ ⊕ W₂, if every vector in W₁ + W₂ can be written uniquely as the sum of a vector in W₁ and a vector in W₂. Equivalently, W₁ + W₂ is a direct sum if and only if W₁ ∩ W₂ = {0}.

Example

In ℝ³, let W₁ be the xy-plane and W₂ be the z-axis. Then W₁ ∩ W₂ = {(0, 0, 0)}, and W₁ + W₂ = ℝ³. Thus, ℝ³ = W₁ ⊕ W₂.

Quotient Spaces

Given a vector space V and a subspace W, one can construct a new vector space called the quotient space, denoted as V/ W. The elements of V/ W are the cosets of W in V.

Cosets

A coset of W in V is a set of the form

v + W = {v + w | w ∈ W}

where v is a vector in V. Intuitively, a coset is a "translation" of the subspace W by the vector v.

Quotient Space

The quotient space V/ W is the set of all cosets of W in V:

V/ W = {v + W | v ∈ V}

The operations of vector addition and scalar multiplication are defined on V/ W as follows:

• (u + W) + (v + W) = (u + v) + W
• c (u + W) = (cu) + W

With these operations, V/ W is a vector space.

Dimension of Quotient Space

If V is finite-dimensional, then the dimension of V/ W is given by:

dim(V/ W) = dim(V) - dim(W)

Example

Let V = ℝ² and let W be the subspace spanned by (1, 0), i.e., the x-axis. Then the cosets of W in V are lines parallel to the x-axis. The quotient space V/ W can be thought of as the set of all these lines. In this case, V/ W is isomorphic to ℝ, the y-axis. We have dim(ℝ²) = 2 and dim(W) = 1, so dim(ℝ²/ W) = 2 - 1 = 1 = dim(ℝ).

Applications of Subspaces

Subspaces play a crucial role in various areas of mathematics, physics, and engineering.

Linear Transformations

The kernel and image of a linear transformation are important examples of subspaces.

• The kernel of a linear transformation T: V → W is the set of all vectors in V that are mapped to the zero vector in W:

  ker(T) = {v ∈ V | T(v) = 0}

  The kernel of T is a subspace of V.

• The image of a linear transformation T: V → W is the set of all vectors in W that are the image of some vector in V:

  im(T) = {T(v) | v ∈ V}

  The image of T is a subspace of W.

The dimensions of the kernel and image are related by the Rank-Nullity Theorem:

dim(ker(T)) + dim(im(T)) = dim(V)

Solving Systems of Linear Equations

The set of solutions to a homogeneous system of linear equations forms a subspace. Consider the system Ax = 0, where A is a matrix and x is a vector. The set of all vectors x that satisfy this equation is a subspace of the vector space in which x lies. This subspace is called the null space or kernel of the matrix A.

Eigenvectors and Eigenspaces

In linear algebra, an eigenvector of a linear transformation is a non-zero vector that, when the transformation is applied, only changes by a scalar factor. This factor is called the eigenvalue.

For a linear transformation T: V → V, a vector v is an eigenvector with eigenvalue λ if

T(v) = λv

The set of all eigenvectors corresponding to a particular eigenvalue λ, together with the zero vector, forms a subspace of V called the eigenspace corresponding to λ.

Eigenspaces are essential in understanding the behavior of linear transformations and are used in various applications, such as solving differential equations and analyzing dynamical systems.

Computer Graphics

Subspaces are used in computer graphics to represent and manipulate objects in 3D space. For example, affine subspaces are used to represent lines, planes, and other geometric primitives. Linear transformations, such as rotations and scalings, can be represented as matrices that act on these subspaces.

Data Analysis

In data analysis and machine learning, subspaces are used in dimensionality reduction techniques such as Principal Component Analysis (PCA). PCA seeks to find a lower-dimensional subspace that captures the most important information in a dataset. The principal components, which are the eigenvectors of the covariance matrix of the data, form a basis for this subspace.

Conclusion

The concept of a subspace is a cornerstone of linear algebra, providing a framework for understanding the structure of vector spaces and linear transformations. Subspaces are subsets of vector spaces that satisfy the axioms of a vector space themselves, ensuring closure under addition and scalar multiplication, and containing the zero vector. Through examples in various vector spaces like ℝ², ℝ³, and polynomial spaces, we've seen how subspaces manifest in diverse forms, from lines and planes through the origin to sets of polynomials with specific properties. Understanding the relationship between subspaces, span, linear independence, and basis is crucial for advanced topics such as quotient spaces and direct sums. The applications of subspaces are vast and impactful, ranging from solving linear equations and understanding linear transformations to enhancing computer graphics and enabling data analysis techniques. By grasping the fundamental principles of subspaces, one gains a deeper insight into the elegance and power of linear algebra.