Glossary

1.1 Vector Spaces and Subspaces

3 min read•july 22, 2024

Vector spaces are the foundation of linear algebra, providing a framework for understanding mathematical structures. They're sets with special rules for adding and scaling elements, allowing us to work with complex systems in a unified way.

From real numbers to polynomials, vector spaces pop up everywhere in math and physics. By mastering their properties and concepts like linear independence and , we gain powerful tools for solving problems in many fields.

Vector Spaces

Properties of vector spaces

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  • Set VV equipped with and operations
  • Vector addition combines two vectors u,v∈Vu, v \in V to produce another vector u+v∈Vu + v \in V ( property)
  • Scalar multiplication scales a vector v∈Vv \in V by a scalar cc to produce another vector cv∈Vcv \in V (closure property)
  • Vector addition is associative (u+v)+w=u+(v+w)(u + v) + w = u + (v + w) and commutative u+v=v+uu + v = v + u for all u,v,w∈Vu, v, w \in V
  • Additive identity element 0∈V0 \in V satisfies v+0=vv + 0 = v for all v∈Vv \in V
  • Additive inverse element −v∈V-v \in V satisfies v+(−v)=0v + (-v) = 0 for every v∈Vv \in V
  • Multiplicative identity scalar 11 satisfies 1v=v1v = v for all v∈Vv \in V
  • Scalar multiplication distributes over vector addition c(u+v)=cu+cvc(u + v) = cu + cv and field addition (a+b)v=av+bv(a + b)v = av + bv for all u,v∈Vu, v \in V and scalars a,b,ca, b, c
  • Scalar multiplication is compatible with field multiplication (ab)v=a(bv)(ab)v = a(bv) for all v∈Vv \in V and scalars a,ba, b

Identification of vector spaces

  • Verify closure under vector addition and scalar multiplication
  • Check associativity and commutativity of vector addition
  • Confirm existence of additive identity and inverses
  • Verify distributivity of scalar multiplication over vector addition and field addition
  • Ensure compatibility of scalar multiplication with field multiplication
  • Common examples: Rn\mathbb{R}^n with standard operations, polynomials with real coefficients, continuous functions on a closed interval

Concepts in vector spaces

  • Linear independence: set of vectors {v1,v2,…,vn}\{v_1, v_2, \ldots, v_n\} is if the equation c1v1+c2v2+…+cnvn=0c_1v_1 + c_2v_2 + \ldots + c_nv_n = 0 has only the trivial solution c1=c2=…=cn=0c_1 = c_2 = \ldots = c_n = 0, otherwise
  • Span: set of all linear combinations of vectors {v1,v2,…,vn}\{v_1, v_2, \ldots, v_n\}, denoted span(v1,v2,…,vn)={c1v1+c2v2+…+cnvn:c1,c2,…,cn∈R}\text{span}(v_1, v_2, \ldots, v_n) = \{c_1v_1 + c_2v_2 + \ldots + c_nv_n : c_1, c_2, \ldots, c_n \in \mathbb{R}\}
  • Basis: linearly independent set of vectors that spans the vector space VV, number of basis vectors equals of VV, every vector in VV uniquely expressed as a of basis vectors

Subspaces and their properties

  • WW of vector space VV is a subset of VV that is itself a vector space under the same operations as VV
  • Properties: zero vector 0∈W0 \in W, closure under vector addition (u,v∈W  ⟹  u+v∈Wu, v \in W \implies u + v \in W) and scalar multiplication (v∈W,c∈R  ⟹  cv∈Wv \in W, c \in \mathbb{R} \implies cv \in W)
  • Examples: zero vector space {0}\{0\} is a subspace of any vector space, lines and planes through origin in R3\mathbb{R}^3, polynomials of degree at most nn form a subspace of all polynomials

Key Terms to Review (18)

Basis: In the context of vector spaces, a basis is a set of linearly independent vectors that spans the entire vector space. This means that any vector in the space can be expressed as a unique linear combination of the basis vectors. Understanding the concept of a basis is crucial for working with vector spaces and subspaces since it allows for the simplification of complex problems by breaking them down into fundamental components.
Closure: Closure refers to a property of a set in which the result of applying a specific operation to elements of that set always produces an element that is also within the same set. This concept is crucial in understanding vector spaces and subspaces, as it helps define how operations like addition and scalar multiplication behave within these structures. When a set is closed under an operation, it implies that performing that operation on elements of the set doesn't lead to results outside of it, maintaining the integrity of the mathematical framework.
Dimension: Dimension refers to the number of coordinates needed to uniquely specify a point in a given space. In the context of vector spaces, it indicates the maximum number of linearly independent vectors that can exist in that space, which helps define its structure. Understanding dimension is crucial when analyzing both vector spaces and linear transformations, as it reveals the relationships and capabilities of these mathematical constructs.
Dimension Theorem: The Dimension Theorem states that the dimension of a vector space is defined as the number of vectors in a basis for that space. This concept highlights the relationship between linear independence, spanning sets, and the structure of vector spaces, providing essential insights into how dimensions relate to subspaces and their respective bases.
Euclidean Space: Euclidean space refers to a mathematical construct that describes a flat, infinite space characterized by the familiar geometric properties established by Euclid. It serves as the fundamental setting for geometry, where points, lines, and shapes are defined, and extends into various dimensions, allowing for a rigorous exploration of concepts like distance and angles. This notion is crucial for understanding vector spaces, transformations, and inner products, as it provides the groundwork upon which these concepts are built.
Finite-dimensional vector space: A finite-dimensional vector space is a vector space that has a finite basis, which means it can be spanned by a finite number of vectors. This characteristic allows for a clear structure in the space, enabling operations like addition and scalar multiplication to be performed easily. In such spaces, the dimension is defined as the number of vectors in the basis, and this dimension provides essential information about the space's properties and behavior.
Function Space: A function space is a set of functions that share common properties and structure, often forming a vector space themselves. Function spaces allow for the application of linear algebra techniques to analyze and manipulate functions, such as addition and scalar multiplication, which are key operations in vector spaces. They provide a framework for studying the behavior of functions in various mathematical contexts, including differential equations and approximation theory.
Hilbert Space: A Hilbert space is a complete inner product space that provides the framework for much of modern mathematical physics, particularly in quantum mechanics. It extends the concept of finite-dimensional Euclidean spaces to infinite dimensions and supports the geometric interpretation of quantum states and their evolution. This makes Hilbert spaces essential for understanding both classical vector spaces and orthogonality concepts in quantum theory, as well as specific applications like the quantum harmonic oscillator.
Infinite-Dimensional Vector Space: An infinite-dimensional vector space is a vector space that has an infinite basis, meaning it cannot be spanned by a finite number of vectors. This type of space extends the concept of finite-dimensional vector spaces, allowing for the representation of more complex structures, such as functions and sequences. Infinite-dimensional spaces are crucial in various fields like functional analysis, quantum mechanics, and differential equations, where they provide a framework for understanding systems with infinitely many degrees of freedom.
Linear Combination: A linear combination is an expression formed by multiplying each vector in a set by a corresponding scalar and then adding the results together. This concept is central to understanding how vectors can be combined to create new vectors within a vector space, emphasizing the importance of scalars and the properties of addition in defining relationships among vectors.
Linearly Dependent: Linearly dependent refers to a situation in which a set of vectors can be expressed as a linear combination of each other, meaning at least one vector in the set can be written as a sum of scalar multiples of the others. This concept is crucial when analyzing vector spaces, as it helps identify relationships among vectors and determines the dimensionality of a space. Understanding linear dependence is key for recognizing whether a collection of vectors spans a vector space or if they merely represent redundant information.
Linearly Independent: Linearly independent refers to a set of vectors in a vector space where no vector can be expressed as a linear combination of the others. This means that each vector adds a unique dimension to the space, and the only way to create the zero vector from this set is by scaling all vectors by zero. The concept of linear independence is crucial because it helps to determine the basis of a vector space, which is essential for understanding its structure and dimensionality.
Rank-Nullity Theorem: The rank-nullity theorem states that for any linear transformation between two finite-dimensional vector spaces, the sum of the rank and the nullity of the transformation equals the dimension of the domain. This fundamental concept highlights the relationship between the dimensions of image and kernel, helping to understand the structure of linear transformations and matrices.
Scalar Multiplication: Scalar multiplication is the operation of multiplying a vector by a scalar, which results in a new vector that is scaled in magnitude but retains its direction. This operation is fundamental in vector spaces, as it allows for the stretching or compressing of vectors while maintaining their directionality, thereby forming a core component of linear transformations and vector operations.
State Space: State space refers to the set of all possible states or configurations that a system can occupy at any given time. This concept is crucial in various fields, as it allows for the organization and representation of complex systems, enabling the analysis of their dynamics and behaviors. By providing a structured framework, state space facilitates the understanding of how systems evolve over time, whether through transformations in a vector space or transitions in probabilistic processes.
Subspace: A subspace is a subset of a vector space that is also a vector space itself, meaning it must satisfy specific conditions such as containing the zero vector, being closed under vector addition, and being closed under scalar multiplication. Understanding subspaces is crucial because they help in analyzing and simplifying complex vector spaces, allowing for the identification of linear combinations and dependencies among vectors.
Vector Addition: Vector addition is the mathematical process of combining two or more vectors to produce a resultant vector. This process follows specific rules, such as the triangle and parallelogram laws, which help to visualize and compute the resulting vector based on the magnitudes and directions of the original vectors. Understanding vector addition is crucial for analyzing physical systems in various fields, as it enables the description of quantities that have both magnitude and direction.
Vector Space: A vector space is a mathematical structure formed by a collection of vectors, which can be added together and multiplied by scalars while satisfying specific properties. This concept allows for operations that combine vectors, enabling the study of linear combinations, transformations, and more in various fields such as physics and engineering. Vector spaces provide a framework for understanding solutions to linear equations and are foundational in various branches of mathematics.
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