Glossary

Vector spaces are the foundation of linear algebra, combining vectors with addition and scalar multiplication. These spaces follow specific rules, ensuring operations stay within the space and behave predictably, which is crucial for mathematical analysis.

Understanding vector spaces is key to grasping more complex concepts in linear algebra. From identifying valid spaces to exploring subspaces and bases, these ideas form the building blocks for studying linear transformations, eigenvalues, and beyond.

Vector Space Fundamentals

Properties of vector spaces

Top images from around the web for Properties of vector spaces

  • Vector space - Wikipedia View original

    Is this image relevant?

  • linear algebra - In Exercises3–12, determine whether each set equipped with the given operations ... View original

    Is this image relevant?

  • Vector space - Wikipedia View original

    Is this image relevant?

1 of 3

Top images from around the web for Properties of vector spaces

  • Vector space - Wikipedia View original

    Is this image relevant?

  • linear algebra - In Exercises3–12, determine whether each set equipped with the given operations ... View original

    Is this image relevant?

  • Vector space - Wikipedia View original

    Is this image relevant?

1 of 3
  • combines set of elements (vectors) with vector addition and scalar multiplication operations
  • Vector space axioms ensure closure, , , identity elements, inverses, and
  • guarantees sum of vectors remains in space
  • ensures scaled vectors stay in space
  • Commutativity of addition allows vector order switching (u+v=v+u\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u})
  • Associativity of addition permits regrouping ((u+v)+w=u+(v+w)(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}))
  • serves as additive identity (v+0=v\mathbf{v} + \mathbf{0} = \mathbf{v})
  • enables vector negation (v+(−v)=0\mathbf{v} + (-\mathbf{v}) = \mathbf{0})
  • Distributivity rules link scalar multiplication with addition
  • preserves vectors (1v=v1\mathbf{v} = \mathbf{v})
  • Common vector spaces include Rn\mathbb{R}^n, complex numbers, and polynomial spaces

Identification of vector spaces

  • Proving vector space requires verifying closure and all axioms
  • Steps to prove vector space:
    1. Check closure under addition and scalar multiplication
    2. Verify all vector space axioms hold
  • Non-vector space examples: positive real numbers (no additive inverse), integers (not closed under scalar multiplication)
  • Counterexample technique finds single case violating an axiom
  • Scalars must form a field (real or complex numbers) for vector space properties

Subspaces and their conditions

  • : non-empty subset of vector space, closed under operations
  • Subspace test ensures subset contains zero vector and closed under linear combinations
  • Examples: planes through origin in R3\mathbb{R}^3, null space of matrix
  • Trivial subspaces include zero subspace and entire vector space
  • Intersection of subspaces always forms a subspace
  • Union of subspaces generally not a subspace (except when one contains the other)

Basis and dimension in vector spaces

  • Linear independence: no vector expressible as of others
  • Spanning set generates entire vector space through linear combinations
  • Basis: linearly independent spanning set, provides unique vector representation
  • Standard basis for Rn\mathbb{R}^n: unit vectors along coordinate axes
  • Dimension equals number of vectors in basis (invariant for given space)
  • Finite-dimensional spaces have finite basis (Euclidean spaces)
  • Infinite-dimensional spaces lack finite basis (function spaces)
  • Finding basis: eliminate dependent vectors from spanning set or extend linearly independent set
  • Change of basis uses transition matrices to convert between coordinate representations
  • Coordinate vectors express elements relative to chosen basis

Key Terms to Review (18)

Additive Inverse: The additive inverse of a number is another number that, when added to the original number, results in zero. This concept is crucial in understanding vector spaces because it highlights the importance of zero as a neutral element and emphasizes the structure of vector addition, where each vector has a corresponding additive inverse within the space.
Associativity: Associativity is a fundamental property in mathematics that states the way in which numbers or operations are grouped does not affect the final result. This concept is crucial in understanding algebraic structures, especially when dealing with vector spaces and group theory, as it ensures that operations can be rearranged without changing outcomes, facilitating simplification and calculation.
Basis Theorem: The Basis Theorem states that every vector space has a basis, which is a set of linearly independent vectors that spans the entire space. This concept is fundamental in understanding the structure of vector spaces and allows for the representation of any vector in the space as a unique linear combination of the basis vectors. The theorem emphasizes the importance of bases in connecting different vector spaces and facilitating transformations between them.
Basis Vectors: Basis vectors are a set of linearly independent vectors that span a vector space, meaning any vector in that space can be expressed as a linear combination of these basis vectors. They provide a reference framework for understanding and describing the properties of the vector space, and play a critical role in defining coordinates and performing operations within that space. In different contexts, such as classical mechanics and quantum mechanics, basis vectors help bridge abstract mathematical concepts with physical interpretations.
Closure under addition: Closure under addition refers to a property of a set in which the sum of any two elements in the set is also an element of that same set. This concept is essential in understanding vector spaces and subspaces, as it ensures that when you add two vectors together, the resulting vector remains within the defined space. It highlights the internal consistency of vector operations, which is fundamental for the structure and behavior of mathematical entities in linear algebra.
Closure under scalar multiplication: Closure under scalar multiplication is a property of a set that indicates if you take any vector in that set and multiply it by a scalar (a real number), the resulting vector will also be in the same set. This property is crucial for defining vector spaces and their subspaces, ensuring that operations within these spaces remain consistent and valid.
Commutativity: Commutativity is a fundamental property in mathematics that states the order in which two elements are combined does not affect the result. In the context of addition and multiplication, for example, this means that for any two elements a and b, it holds true that a + b = b + a and a * b = b * a. This property is essential for simplifying expressions and operations within various mathematical structures, including vector spaces and subspaces.
Dimension Theorem: The Dimension Theorem states that the dimension of a finite-dimensional vector space is equal to the maximum number of linearly independent vectors that can exist within that space. This concept highlights the relationship between the size of a vector space and the vectors that span it, allowing for a deeper understanding of the structure and properties of vector spaces and their subspaces.
Distributivity: Distributivity is a fundamental property of mathematical operations that describes how multiplication interacts with addition or subtraction. In simple terms, it states that when you multiply a sum by a number, you can distribute the multiplication across each term of the sum. This property is essential for simplifying expressions and solving equations within vector spaces and subspaces, as it helps establish how linear combinations work.
Euclidean Space: Euclidean space is a mathematical construct that defines a flat, two-dimensional or three-dimensional space characterized by the concepts of points, lines, and planes. In this framework, geometric properties can be described using coordinate systems, allowing for the representation of vectors and geometric shapes. This foundational concept is essential for understanding vector spaces, subspaces, and inner product spaces, which form the basis for further mathematical analysis and applications.
Finite-dimensional vector space: A finite-dimensional vector space is a vector space that has a finite basis, meaning that it can be spanned by a finite number of vectors. This characteristic allows for a straightforward understanding of the space's structure, as any vector in this space can be expressed as a linear combination of these basis vectors. In relation to linear transformations and matrices, finite-dimensional vector spaces play a crucial role because the transformations can be represented using matrices that map between these spaces.
Function Space: A function space is a collection of functions that share a common domain and codomain, forming a vector space when considering functions as vectors. This concept is essential in mathematical analysis, where function spaces are often equipped with additional structures like norms or inner products, allowing for the study of convergence, continuity, and orthogonality within these spaces.
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 concept extends the idea of vector spaces beyond the familiar finite dimensions, allowing for more complex structures that arise in various mathematical contexts, such as functional analysis and quantum mechanics. These spaces often require different techniques for analysis and manipulation compared to finite-dimensional spaces.
Linear combination: A linear combination refers to an expression formed by multiplying each vector in a set by a corresponding scalar and then summing the results. This concept is fundamental in understanding vector spaces, as it helps define how vectors can be combined to produce new vectors, revealing the structure and relationships within those spaces.
Scalar multiplication identity: The scalar multiplication identity refers to the property of a vector space where multiplying any vector by the scalar value of 1 yields the original vector unchanged. This property is essential in understanding how vectors interact within a vector space, showcasing that the scalar 1 acts as a neutral element in scalar multiplication.
Subspace: A subspace is a subset of a vector space that is itself a vector space, meaning it must be closed under vector addition and scalar multiplication. This concept connects deeply with vector spaces and linear transformations because understanding how subspaces behave under these operations can help in analyzing more complex structures and transformations within larger vector spaces.
Vector Space: A vector space is a mathematical structure formed by a collection of vectors, which can be added together and multiplied by scalars, satisfying specific axioms. These axioms govern how vectors can be combined and manipulated, leading to a wide variety of applications in different fields, such as physics and engineering. The idea of a vector space is foundational in linear algebra and serves as the basis for understanding more complex structures like subspaces and linear transformations.
Zero Vector: The zero vector is a unique vector in a vector space that has all its components equal to zero. It serves as the additive identity in the context of vector addition, meaning that when any vector is added to the zero vector, the original vector remains unchanged. This property makes the zero vector essential in establishing the structure and behavior of vector spaces and subspaces.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide