Glossary

3.3 Basis and dimension

4 min read•august 16, 2024

and are key concepts in vector spaces. They help us understand the structure and size of these spaces. A basis is a set of vectors that can generate any vector in the space, while dimension tells us how many vectors we need in a basis.

These ideas are crucial for analyzing vector spaces and subspaces. They let us represent vectors efficiently, solve systems of equations, and understand the relationships between different spaces. Mastering basis and dimension is essential for tackling more advanced topics in linear algebra.

Basis and Dimension of Vector Spaces

Fundamental Concepts of Basis and Dimension

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  • A basis of a consists of vectors that span the entire vector space
  • The dimension of a vector space equals the number of vectors in any basis of that space
  • A basis must satisfy two key properties
    • Linear independence prevents expressing any basis vector as a linear combination of other basis vectors
    • allows expressing every vector in the space as a linear combination of basis vectors
  • for R^n contains n vectors, each with a 1 in one position and 0s elsewhere (e1=(1,0,0,…,0)e_1 = (1,0,0,\ldots,0), e2=(0,1,0,…,0)e_2 = (0,1,0,\ldots,0), etc.)
  • Infinite-dimensional vector spaces have bases but require advanced mathematical techniques to define and work with (function spaces)

Properties and Applications of Basis and Dimension

  • The dimension of a vector space remains invariant regardless of the chosen basis
  • Dimension of R^n equals n, while dimension of n×n matrices space equals n^2
  • dimension always less than or equal to the containing space dimension
  • ###-Nullity_Theorem_0### states dim⁡(V)=\rank(T)+\nullity(T)\dim(V) = \rank(T) + \nullity(T) for T: V → W
  • Dimension of polynomial space P_n (degree at most n) equals n+1
  • Infinite-dimensional spaces (function spaces) require advanced mathematical frameworks like cardinal numbers for dimension concept

Finding a Basis for a Vector Space

Methods for Constructing Bases

  • Gram-Schmidt process converts linearly independent vectors into orthogonal or
  • Row reduction (Gaussian elimination) finds basis for column space or row space of a matrix
  • Null space of coefficient matrix forms basis for solution space of system of equations
  • Eliminate linear dependencies among given vectors to find basis for subspace defined by span of vectors
  • Construct basis for polynomial vector spaces using monomials of appropriate degrees (1, x, x^2, etc.)
  • Identify fundamental elements generating the space and prove their linear independence for abstract vector spaces

Applications and Properties of Bases

  • Rank of a matrix equals number of vectors in basis for its column space or row space
  • Basis for subspace defined by system of equations derived from null space of coefficient matrix
  • Polynomial basis construction involves selecting appropriate degree monomials (constant term, linear term, quadratic term, etc.)
  • Abstract vector space basis identification requires proving linear independence of generating elements

Dimension of a Vector Space

Calculating and Understanding Dimension

  • Dimension equals number of vectors in any basis for finite-dimensional vector spaces
  • Subspace dimension always less than or equal to containing space dimension
  • Rank-nullity theorem: dim⁡(V)=\rank(T)+\nullity(T)\dim(V) = \rank(T) + \nullity(T) for linear transformation T: V → W
  • Dimension of polynomial space P_n (degree at most n) equals n+1
  • Infinite-dimensional spaces (function spaces) require advanced mathematical frameworks for dimension concept

Examples and Applications of Dimension

  • R^3 has dimension 3, with standard basis vectors (1,0,0)(1,0,0), (0,1,0)(0,1,0), and (0,0,1)(0,0,1)
  • 2×2 matrix space has dimension 4, with basis (1000)\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}, (0100)\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}, (0010)\begin{pmatrix}0 & 0 \\ 1 & 0\end{pmatrix}, (0001)\begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}
  • Polynomial space P_2 (quadratic polynomials) has dimension 3, with basis {1,x,x2}\{1, x, x^2\}
  • Function space C[0,1] (continuous functions on [0,1]) infinite-dimensional, requiring advanced concepts

Linear Combinations of Basis Vectors

Expressing Vectors Using Basis

  • Any vector in a vector space uniquely expressed as linear combination of basis vectors
  • Coefficients in linear combination called coordinates of vector with respect to given basis
  • For basis {v1,v2,…,vn}\{v_1, v_2, \ldots, v_n\} of vector space V, any vector w in V written as w=c1v1+c2v2+…+cnvnw = c_1v_1 + c_2v_2 + \ldots + c_nv_n for unique scalars c1,c2,…,cnc_1, c_2, \ldots, c_n
  • Finding coefficients often involves solving system of linear equations
  • R^n with standard basis has vector coordinates simply as its components
  • Change of basis formula converts coordinates from one basis to another

Applications and Examples of Linear Combinations

  • In R^3, vector (2,3,-1) expressed as 2(1,0,0)+3(0,1,0)+(−1)(0,0,1)2(1,0,0) + 3(0,1,0) + (-1)(0,0,1) using standard basis
  • Polynomial p(x)=3x2−2x+1p(x) = 3x^2 - 2x + 1 in P_2 expressed as 3(x2)+(−2)(x)+1(1)3(x^2) + (-2)(x) + 1(1) using standard polynomial basis
  • Expressing vectors as linear combinations of basis vectors crucial for efficient storage and manipulation of high-dimensional data in computational applications
  • In quantum mechanics, state vectors expressed as linear combinations of basis states to represent superposition

Key Terms to Review (18)

Basis: A basis is a set of vectors in a vector space that are linearly independent and span the entire space, meaning any vector in that space can be expressed as a linear combination of the basis vectors. The concept of basis is essential for understanding the structure and dimensionality of vector spaces, as well as the transformations that can be applied to them.
Dimension: Dimension refers to the number of independent directions in a space, which is fundamentally connected to the concept of vector spaces. It tells us how many vectors are needed to form a basis for that space, indicating how many coordinates are required to represent points within it. This idea is crucial in understanding subspaces, as they can have dimensions that are less than or equal to the dimension of the entire space, and influences the properties of vector spaces themselves, including their representation using scalars, vectors, and matrices.
Dimension Theorem: The dimension theorem states that in a finite-dimensional vector space, the dimension can be expressed in terms of the rank and nullity of a linear transformation. This relationship is significant as it ties together several key concepts, including how many linearly independent vectors can span a space (basis), and the structure of vector spaces through their properties. Understanding this theorem helps clarify the relationships between different dimensions associated with transformations between vector spaces.
Dimensionality Reduction: Dimensionality reduction is a process used to reduce the number of random variables under consideration, obtaining a set of principal variables. It simplifies models, making them easier to interpret and visualize, while retaining important information from the data. This technique connects with various linear algebra concepts, allowing for the transformation and representation of data in lower dimensions without significant loss of information.
Embedding: Embedding refers to a mathematical representation of one space within another, often used to illustrate how elements from one vector space can be mapped into a higher-dimensional vector space. This concept is crucial as it allows for the transformation and visualization of data, aiding in dimensionality reduction and facilitating operations like interpolation and extrapolation.
Feature Space: Feature space is a multi-dimensional space where each dimension represents a feature or attribute of the data used in machine learning and data analysis. It serves as a geometric representation of the data points, where each point corresponds to a unique instance of the data defined by its features. Understanding feature space is crucial for determining how algorithms learn from data and how different features influence the outcome of models.
Isomorphism: Isomorphism refers to a mapping between two structures that preserves the operations and relations of those structures, indicating they are fundamentally the same in terms of their properties. In linear algebra, this concept highlights the relationship between vector spaces, revealing when two spaces can be considered equivalent by their dimensions and bases. It also plays a crucial role in understanding matrix inverses and determinants, showing how transformations can be applied while maintaining essential characteristics.
Linear Transformation: A linear transformation is a mathematical function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. This means that if you have a linear transformation, it will take a vector and either stretch, rotate, or reflect it in a way that keeps the relationships between vectors intact. Understanding how these transformations work is crucial in many areas like eigendecomposition, matrix representation, and solving problems in data science.
Linearly independent: Linearly independent refers to a set of vectors in a vector space that do not express any vector in the set as a linear combination of the others. This means that no vector can be formed by adding or scaling the others, indicating that each vector contributes a unique direction in the space. This concept is crucial in understanding how to construct bases and determine dimensions of vector spaces.
Nullity: Nullity is the dimension of the null space of a linear transformation or matrix, representing the number of linearly independent solutions to the homogeneous equation associated with that transformation. It measures the extent to which a linear transformation fails to be injective, revealing important insights about the relationships among vectors in vector spaces and their mappings.
Orthonormal Basis: An orthonormal basis is a set of vectors in a vector space that are both orthogonal and normalized, meaning each vector has a length of one, and every pair of different vectors in the set is perpendicular to each other. This concept is crucial for simplifying calculations in linear algebra, especially when working with projections and transformations, as it enables a clear structure for representing data and functions in high-dimensional spaces.
Principal Component Analysis: Principal Component Analysis (PCA) is a statistical technique used to simplify data by reducing its dimensionality while retaining the most important features. By transforming a large set of variables into a smaller set of uncorrelated variables called principal components, PCA helps uncover patterns and structures within the data, making it easier to visualize and analyze.
Rank: In linear algebra, rank is the dimension of the column space of a matrix, which represents the maximum number of linearly independent column vectors in that matrix. It provides insight into the solution space of linear systems, helps understand transformations, and plays a crucial role in determining properties like consistency and dimensionality of vector spaces.
Rank-Nullity Theorem: The Rank-Nullity Theorem states that for any linear transformation represented by a matrix, the sum of the rank and the nullity of the transformation equals the number of columns of the matrix. This theorem connects key concepts such as linear transformations, matrix representation, and subspaces, providing insight into how the dimensions of various vector spaces are related to each other.
Spanning Set: A spanning set is a collection of vectors in a vector space that can be combined through linear combinations to produce every vector in that space. This means that if you take any vector in the space, you can express it as a sum of the vectors in the spanning set, multiplied by some coefficients. Understanding spanning sets is crucial for grasping the concepts of basis and dimension, as they help define the entire structure of the vector space and determine its dimensionality.
Standard Basis: The standard basis is a set of vectors that forms a basis for a vector space, where each vector in the set has one component equal to 1 and all other components equal to 0. This collection of vectors provides a simple way to express any vector in the space as a linear combination, making it easy to understand concepts like basis and dimension.
Subspace: A subspace is a set of vectors that forms a vector space within a larger vector space, satisfying the same axioms and properties as the original space. It must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication. Understanding subspaces helps in grasping important concepts like orthogonality, basis, dimension, and the structure of vector spaces.
Vector Space: A vector space is a mathematical structure formed by a collection of vectors that can be added together and multiplied by scalars, adhering to specific rules. It is fundamental in understanding linear combinations, linear independence, and spans, which are crucial for various applications in linear transformations, subspaces, and dimensional analysis.
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