Glossary

1.5 Basis and dimension of a vector space

3 min read•august 16, 2024

Vector spaces are the foundation of linear algebra. Bases and help us understand their structure. A is a set of vectors that spans the space and is linearly independent. It's like a skeleton that defines the space's shape.

Dimension tells us how many vectors are in a basis. It's a key property of vector spaces, helping us compare and classify them. Understanding bases and dimension is crucial for solving linear systems and analyzing transformations between spaces.

Basis of a Vector Space

Definition and Key Properties

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  • A basis comprises a linearly independent subset of vectors that spans the entire vector space
  • Multiple sets of vectors can form a basis for a given vector space
  • Express every vector in the space as a unique linear combination of basis vectors
  • Finite-dimensional vector spaces always have a finite number of basis vectors
  • Removing any vector from a basis results in a set no longer spanning the space
  • Basis provides a allowing unique representation of vectors

Examples and Applications

  • for R3\mathbb{R}^3: (1,0,0),(0,1,0),(0,0,1)(1,0,0), (0,1,0), (0,0,1)
  • Polynomial basis for P2P_2: {1,x,x2}\{1, x, x^2\}
  • Fourier basis for periodic functions: {1,sin⁡(x),cos⁡(x),sin⁡(2x),cos⁡(2x),...}\{1, \sin(x), \cos(x), \sin(2x), \cos(2x), ...\}
  • Basis for matrix space M2x2M_{2x2}: (1000),(0100),(0010),(0001)\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}

Basis Cardinality

Proof Concepts and Techniques

  • Utilize and spanning properties of bases in the proof
  • Apply the Replacement Theorem (Exchange Lemma) to transform one basis into another
  • Maintain linear independence and spanning property during vector replacements
  • Use contradiction to show different cardinalities violate basis definition
  • Demonstrate invariance of basis vector count for a given vector space
  • Establish foundation for vector space dimension concept

Proof Outline and Examples

  • Start with two bases B₁ and B₂ of vector space
  • Assume |B₁| > |B₂| and derive a contradiction
    • Show a linear dependence in B₁ using vectors from B₂
    • Contradiction violates basis definition
  • Repeat assuming |B₂| > |B₁| to show equality
  • Example: Prove standard basis and diagonal matrix basis for M2x2M_{2x2} have same cardinality
  • Application: Prove dimension of PnP_n (polynomials of degree ≤ n) is n+1

Finding a Basis

Methods and Techniques

  • Apply Gram-Schmidt process to create orthogonal or from linearly independent vectors
  • Use Gaussian elimination for null space basis of linear equation systems
  • Identify linearly independent columns for matrix column space basis
  • Construct standard bases using monomials for polynomial vector spaces
  • Eliminate linear dependencies among spanning vectors
  • Employ Steinitz exchange lemma to extend linearly independent set or reduce

Examples and Applications

  • Orthonormalize vectors (1,1,0),(1,0,1),(0,1,1)(1,1,0), (1,0,1), (0,1,1) in R3\mathbb{R}^3 using Gram-Schmidt
  • Find basis for null space of matrix A=(123246)A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \end{pmatrix}
  • Determine column space basis for matrix B=(123011134)B = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 1 & 3 & 4 \end{pmatrix}
  • Construct basis for P3P_3 (polynomials of degree ≤ 3)
  • Use Steinitz exchange to find basis of subspace spanned by (1,1,1),(1,2,3),(2,3,4)(1,1,1), (1,2,3), (2,3,4) in R3\mathbb{R}^3

Dimension of a Vector Space

Definition and Properties

  • Dimension equals number of vectors in any basis of the space
  • Finite-dimensional spaces have non-negative integer dimensions
  • Zero vector space has dimension 0 (empty set basis)
  • Subspace dimension ≤ parent vector space dimension
  • Calculate dimension by finding a basis and counting its vectors
  • relates vector space dimension to range and null space dimensions

Calculation Methods and Examples

  • Determine dimension of Rn\mathbb{R}^n (n)
  • Calculate dimension of PnP_n (n+1)
  • Find dimension of MmxnM_{mxn} (m×n)
  • Compute dimension of solution space for homogeneous system Ax = 0
  • Use rank-nullity theorem to find nullity of T: R4\mathbb{R}^4 → R3\mathbb{R}^3 with rank 2
  • Calculate dimension of span{(1,1,0), (0,1,1), (1,0,1)} in R3\mathbb{R}^3

Key Terms to Review (16)

Basis: A basis is a set of vectors in a vector space that are linearly independent and span the entire space. This means that any vector in the space can be expressed as a linear combination of the basis vectors. Understanding the concept of a basis is crucial because it helps define the structure of a vector space, connecting ideas like linear independence, dimension, and coordinate systems.
Coordinate System: A coordinate system is a mathematical framework that allows for the unique identification of points in space using numerical coordinates. It consists of an origin point and a set of axes, which provide a way to describe the position and orientation of vectors within a vector space. The choice of a coordinate system can significantly impact the representation and analysis of vectors, as different bases can lead to different coordinate representations for the same vector.
Dimension: Dimension is a measure of the number of vectors in a basis of a vector space, reflecting the space's capacity to hold information. It plays a crucial role in understanding the structure of vector spaces, where the dimension indicates the maximum number of linearly independent vectors that can exist within that space. This concept helps in characterizing spaces, determining whether sets of vectors can span them, and understanding how different types of spaces relate to one another.
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.
Functional Analysis: Functional analysis is a branch of mathematical analysis that studies spaces of functions and the linear operators acting upon them. It focuses on understanding properties of function spaces, like their bases and dimensions, as well as the behaviors of operators, especially in terms of self-adjointness and normality. This area of study is crucial for various applications in mathematics and physics, particularly in solving differential equations and understanding quantum mechanics.
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.
Isomorphism: Isomorphism is a mathematical concept that describes a structure-preserving mapping between two algebraic structures, such as vector spaces or groups, indicating that they are essentially the same in terms of their properties and operations. This concept highlights how two different systems can be related in a way that preserves the underlying structure, allowing for insights into their behavior and characteristics.
Linear Independence: Linear independence refers to a set of vectors in which no vector can be expressed as a linear combination of the others. This concept is essential for understanding the structure of vector spaces, as it helps identify which vectors can span a space without redundancy, making them crucial in defining bases and dimensions.
Linear transformation: A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means if you take any two vectors and apply the transformation, the result will be the same as transforming each vector first and then adding them together. It connects to various concepts, showing how different bases interact, how they can change with respect to matrices, and how they impact the underlying structure of vector spaces.
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.
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.
Rank-Nullity Theorem: The Rank-Nullity Theorem states that for any linear transformation from one vector space to another, the sum of the rank (the dimension of the image) and the nullity (the dimension of the kernel) is equal to the dimension of the domain. This theorem helps illustrate relationships between different aspects of vector spaces and linear transformations, linking concepts like subspaces, linear independence, and matrix representations.
Row Reduction: Row reduction is a method used to simplify a matrix into its row echelon form or reduced row echelon form through a series of elementary row operations. This process helps in solving systems of linear equations, finding bases for vector spaces, and determining the rank of a matrix, which are all crucial in understanding vector spaces and linear transformations.
Spanning Set: A spanning set for a vector space is a collection of vectors that, through linear combinations, can generate every vector in that space. This means that any vector within the space can be expressed as a sum of scalar multiples of the vectors in the spanning set. Understanding spanning sets is crucial for exploring subspaces and determining the basis and dimension of a vector space, as they establish the foundational elements that define the entire space.
Standard Basis: The standard basis is a specific set of vectors that provides a reference for all other vectors in a given vector space. In $ ext{R}^n$, the standard basis consists of the unit vectors $ ext{e}_1, ext{e}_2, ..., ext{e}_n$, where each vector has a 1 in one coordinate and 0s in all others. This basis is crucial for understanding how vectors can be expressed in terms of coordinates and how transformations between different bases can occur.
V: In the context of vector spaces, 'v' typically represents a vector, which is an element of a vector space. Vectors can be thought of as ordered tuples of numbers that represent points in space, or as arrows that have both direction and magnitude. They are fundamental building blocks in linear algebra, and understanding their properties is essential for grasping the concepts of basis and dimension within vector spaces.
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