Glossary

1.1 Definition and examples of vector spaces

3 min read•august 16, 2024

Vector spaces form the foundation of linear algebra, combining and . These operations follow specific rules, ensuring consistent behavior across various mathematical structures.

From Euclidean spaces to function spaces, vector spaces appear in many forms. Understanding their axioms and examples is crucial for grasping more advanced concepts in linear algebra and its applications.

Axioms of Vector Spaces

Vector Addition and Scalar Multiplication

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  • V over field F combines two operations
    • Vector addition (binary operation on V)
    • Scalar multiplication (operation between F and V elements)
  • Vector addition exhibits key properties
    • Commutative u+v=v+u\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}
    • Associative (u+v)+w=u+(v+w)(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})
    • Identity element () 0+v=v\mathbf{0} + \mathbf{v} = \mathbf{v}
    • Inverse elements v+(−v)=0\mathbf{v} + (-\mathbf{v}) = \mathbf{0}
  • Scalar multiplication satisfies distributivity and compatibility
    • Distributive over vector addition a(u+v)=au+ava(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}
    • Distributive over scalar addition (a+b)v=av+bv(a + b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}
    • Compatible with field multiplication (ab)v=a(bv)(ab)\mathbf{v} = a(b\mathbf{v})

Uniqueness and Closure Properties

  • Zero vector uniqueness ensures single additive identity
  • Additive inverse uniqueness guarantees v+(−v)=0\mathbf{v} + (-\mathbf{v}) = \mathbf{0} for each vector
  • Multiplicative identity of field (typically 1) acts as scalar multiplication identity
    • 1v=v1\mathbf{v} = \mathbf{v} for all vectors v
  • Closure property maintains vector space integrity
    • Vector addition result always within V
    • Scalar multiplication result always within V

Examples of Vector Spaces

Finite-Dimensional Spaces

  • Euclidean spaces Rⁿ (n ≥ 1) over real numbers
    • Componentwise addition (x1,...,xn)+(y1,...,yn)=(x1+y1,...,xn+yn)(x_1, ..., x_n) + (y_1, ..., y_n) = (x_1 + y_1, ..., x_n + y_n)
    • Scalar multiplication a(x1,...,xn)=(ax1,...,axn)a(x_1, ..., x_n) = (ax_1, ..., ax_n)
  • Polynomial spaces P_n(F) over field F
    • Degree ≤ n polynomials
    • Addition (a0+a1x+...+anxn)+(b0+b1x+...+bnxn)=((a0+b0)+(a1+b1)x+...+(an+bn)xn)(a_0 + a_1x + ... + a_nx^n) + (b_0 + b_1x + ... + b_nx^n) = ((a_0 + b_0) + (a_1 + b_1)x + ... + (a_n + b_n)x^n)
    • Scalar multiplication c(a0+a1x+...+anxn)=(ca0+ca1x+...+canxn)c(a_0 + a_1x + ... + a_nx^n) = (ca_0 + ca_1x + ... + ca_nx^n)
  • Matrix spaces M_m,n(F) over field F
    • m × n matrices
    • Addition [aij]+[bij]=[aij+bij][a_{ij}] + [b_{ij}] = [a_{ij} + b_{ij}]
    • Scalar multiplication c[aij]=[caij]c[a_{ij}] = [ca_{ij}]

Infinite-Dimensional Spaces

  • Continuous function space C[a,b] on interval [a,b]
    • Pointwise addition (f+g)(x)=f(x)+g(x)(f + g)(x) = f(x) + g(x)
    • Scalar multiplication (cf)(x)=cf(x)(cf)(x) = cf(x)
  • Sequence space over field F
    • Infinite-dimensional vectors (a₁, a₂, a₃, ...)
    • Componentwise operations
  • Solution space of homogeneous linear differential equations
    • Linear combinations of fundamental solutions
  • Advanced function spaces
    • L²[a,b] (square-integrable functions)
    • C∞(R) (infinitely differentiable functions)

Verifying Vector Spaces

Axiom Verification Process

  • Systematically check all vector space axioms for given set and operations
  • Verify closure property for vector addition and scalar multiplication
    • Ensure operations result in elements within the set
  • Confirm vector addition properties
    • Commutativity u+v=v+u\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}
    • Associativity (u+v)+w=u+(v+w)(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})
    • Zero vector existence 0+v=v\mathbf{0} + \mathbf{v} = \mathbf{v}
    • Additive inverse existence v+(−v)=0\mathbf{v} + (-\mathbf{v}) = \mathbf{0}
  • Check scalar multiplication properties
    • Distributivity over vector addition a(u+v)=au+ava(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}
    • Distributivity over scalar addition (a+b)v=av+bv(a + b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}
    • Compatibility with field multiplication (ab)v=a(bv)(ab)\mathbf{v} = a(b\mathbf{v})
    • Multiplicative identity 1v=v1\mathbf{v} = \mathbf{v}

Special Considerations and Challenges

  • Identify potential counterexamples violating axioms
    • Non-closure (operation results outside the set)
    • Lack of commutativity or associativity
    • Missing or non-unique zero vector or inverses
  • Prove uniqueness of zero vector and inverse elements
    • Demonstrate 0+v=v\mathbf{0} + \mathbf{v} = \mathbf{v} holds only for one element
    • Show v+(−v)=0\mathbf{v} + (-\mathbf{v}) = \mathbf{0} has unique solution for each v
  • Analyze abstract or non-standard vector spaces carefully
    • Examine operation definitions closely
    • Consider implications of unusual scalar fields (finite fields, complex numbers)
  • Verify scalar multiplication behaves correctly with field operations
    • Check compatibility with addition and multiplication in the scalar field

Key Terms to Review (18)

Basis Theorem: The Basis Theorem states that any vector space has a basis, which is a set of linearly independent vectors that span the entire space. This theorem emphasizes the importance of bases in understanding the structure of vector spaces, allowing for a simplified representation of vectors and their properties. The existence of a basis also leads to the idea that all bases of a vector space have the same cardinality, known as the dimension of the space.
Closure under addition: Closure under addition refers to the property that when you take any two elements from a set and add them together, the result is also an element of that same set. This idea is crucial for defining vector spaces and subspaces, as it ensures that the set remains intact when performing the operation of addition, which is one of the foundational operations in linear algebra. When a set possesses this property, it helps to confirm whether the set can be classified as a vector space or a subspace, and is also significant in understanding how different subspaces can combine through sum and direct sum operations.
Closure under scalar multiplication: Closure under scalar multiplication refers to the property that if you take any vector in a set and multiply it by any scalar, the resulting vector is also in that set. This concept is essential in defining vector spaces and their subspaces, as it ensures that operations on vectors produce other vectors within the same structure, preserving the integrity of the space. When a set satisfies closure under scalar multiplication, it guarantees that the vector space or subspace remains consistent with its defined operations.
Dimension Theorem: The Dimension Theorem states that for any vector space, the dimension is equal to the number of vectors in a basis for that space. This theorem connects crucial concepts like vector spaces, linear independence, and bases, highlighting how these elements interact to determine the size and structure of the space.
Finite-dimensional vector space: A finite-dimensional vector space is a type of vector space that has a finite basis, meaning it can be spanned by a finite number of vectors. This property allows for clear dimensionality, where the dimension is the number of vectors in the basis and provides a structured way to analyze linear combinations, linear independence, and subspaces. Understanding finite-dimensional spaces is crucial for studying the behavior of vectors and linear transformations within a controlled framework.
Hamel Basis: A Hamel basis is a specific type of basis for a vector space, where every vector in the space can be expressed as a finite linear combination of vectors from the basis. This concept is essential in understanding the structure of vector spaces, especially in infinite dimensions, as it helps establish how dimensions are defined and how they can be manipulated within a vector space.
Infinite-dimensional vector space: An infinite-dimensional vector space is a vector space that contains an infinite basis, meaning that there is no finite set of vectors that can span the entire space. This type of space extends the concept of vector spaces beyond the familiar finite dimensions and is crucial in various fields such as functional analysis and quantum mechanics. Infinite-dimensional vector spaces are often associated with functions, sequences, or other objects that cannot be confined to a limited number of dimensions.
Linear Combination: A linear combination is an expression formed by multiplying each vector in a set by a corresponding scalar and then summing the results. This concept is essential for understanding how vectors can be combined to produce new vectors and plays a crucial role in defining vector spaces, determining the structure of subspaces, and assessing linear independence or dependence among vectors.
Orthonormal Basis: An orthonormal basis is a set of vectors in a vector space that are both orthogonal to each other and normalized to have a length of one. This concept is fundamental in understanding the structure of vector spaces and facilitates easier calculations, especially when dealing with projections, transformations, and inner product spaces.
Polynomial Space p_n(f): The polynomial space p_n(f) is the vector space consisting of all polynomials of degree at most n, with coefficients drawn from a field f. This space includes the zero polynomial and allows for operations like addition and scalar multiplication, making it a structured environment for understanding polynomials as vectors in a vector space.
R^n: The term $$\mathbb{R}^n$$ refers to the n-dimensional Euclidean space, consisting of all possible ordered n-tuples of real numbers. It serves as a fundamental example of a vector space, where the vectors are represented as coordinates in this n-dimensional space. Understanding $$\mathbb{R}^n$$ helps in visualizing and working with concepts such as linear combinations, span, and dimensionality within the framework of vector spaces.
Scalar Multiplication: Scalar multiplication is an operation that involves multiplying a vector by a scalar (a real number), which results in a new vector that points in the same or opposite direction depending on the sign of the scalar and has its magnitude scaled accordingly. This operation is fundamental to understanding how vectors behave within vector spaces, as it helps define their structure and properties. It also plays a crucial role when discussing coordinate vectors and changes of basis, as it allows for the transformation and manipulation of vectors within different coordinate systems.
Span: Span refers to the set of all possible linear combinations of a given set of vectors. It represents all the points that can be reached in a vector space through these combinations, effectively capturing the extent of coverage these vectors have within that space. The concept of span connects deeply with understanding vector spaces, the relationships between vectors regarding independence and dependence, how coordinates shift during basis changes, and the creation of orthogonal sets in processes like Gram-Schmidt.
Subspace: A subspace is a subset of a vector space that is itself a vector space under the same operations of addition and scalar multiplication. It must satisfy three conditions: it contains the zero vector, it is closed under vector addition, and it is closed under scalar multiplication. Understanding subspaces is crucial, as they play a significant role in the structure of vector spaces and are foundational for various concepts like dimension and basis.
Unit vector: A unit vector is a vector that has a magnitude of exactly one. This property makes unit vectors particularly useful in mathematics and physics, as they are often used to indicate direction without implying any specific magnitude. In the context of vector spaces, unit vectors can serve as basis vectors, helping to define and represent other vectors through linear combinations.
Vector Addition: Vector addition is the process of combining two or more vectors to produce a new vector. This operation is fundamental in vector spaces, as it allows for the exploration of properties like closure and linear combinations. Understanding vector addition also lays the groundwork for working with coordinate vectors, where it helps in visualizing and manipulating vectors in different bases.
Vector Space: A vector space is a mathematical structure formed by a collection of vectors that can be added together and multiplied by scalars while satisfying specific axioms. These axioms ensure that operations such as vector addition and scalar multiplication are well-defined, leading to rich applications in areas such as geometry and algebra. Understanding vector spaces is crucial for grasping concepts like linear independence, basis, and dimension, all of which play pivotal roles in linear transformations and systems of equations.
Zero vector: The zero vector is a special vector in a vector space that serves as the additive identity, meaning when it is added to any vector, it does not change that vector. This unique vector has all of its components equal to zero, and it plays a critical role in defining the structure of vector spaces, determining linear independence, and forming quotient spaces as well as isomorphism theorems.
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