PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Algebra → Linear Algebra ↓


Vector Spaces


In the study of linear algebra, an important concept is that of vector spaces. Vector spaces provide a framework for working with vectors, which can be thought of as quantities having both magnitude and direction. These fundamental structures are essential in various fields of mathematics, physics, engineering, and computer science.

Definition of vector space

A vector space, or linear space, is a collection of objects called vectors. These vectors can be added together and multiplied by numbers, called scalars. Scalars are often real numbers, but they can also be complex numbers or elements of a specified field (a mathematical term that covers numbers on which certain operations are defined that follow certain rules).

More formally, a vector space V over a field F is a set equipped with two operations: vector addition and scalar multiplication. The vector space must satisfy the ten axioms listed below:

  • Closure under addition: for all u, v in V, the sum u + v is also in V
  • Closure under scalar multiplication: for any scalar c in F and any vector v in V, the product c * v is in V
  • Associative property of addition: For all u, v, w in V, (u + v) + w = u + (v + w)
  • Commutative property of addition: For all u, v in V, u + v = v + u.
  • Identity element of sum: There exists an element 0 in V, called the zero vector, such that v + 0 = v for all v in V
  • Inverse elements of a sum: for every v in V, there exists a -v in V such that v + (-v) = 0.
  • Compatibility of scalar multiplication with field multiplication: for all a, b in F and v in V, (a * b) * v = a * (b * v)
  • Identity element of scalar multiplication: for all v in V, 1 * v = v, where 1 is the multiplicative identity in F
  • Distributive property of scalar multiplication with respect to vector addition: for all a in F and u, v in V, a * (u + v) = a * u + a * v.
  • Distributive property of scalar multiplication with respect to field addition: for all a, b in F and v in V, (a + b) * v = a * v + b * v.

Geometric interpretation of a vector space

Let's look at a vector space in a more geometric sense. Imagine the simple case of a two-dimensional space, usually denoted as R^2. Here the vectors are line segments directed from the origin.

V You u+v

This diagram shows the vectors u, v and u+v in two-dimensional space.

In this simple 2D case, the sum of two vectors u and v can also be found graphically. Place them so that the tail of v is at the head of u, and their sum is the vector from the base of u to the head of v. This is also known mathematically as the "parallelogram rule".

Examples of vector spaces

Now, let's explore different examples of vector spaces:

Example 1: Physical location

The most intuitive vector space might be the 3-dimensional physical space around us, R^3. The basic operations of vector addition and scalar multiplication can be easily visualized here by moving through the space.

Example 2: Polynomial

Consider the set of all polynomials with real coefficients, R[x]. This is a vector space where vector addition is polynomial addition, and scalar multiplication is multiplying each term in the polynomial by the same scalar.

Example 3: Function

The set of all continuous real-valued functions defined on a closed interval is another example. Here, vector summation corresponds to function addition, and scalar multiplication again modifies the 'height' of each function by a scalar.

Subspace

A subspace is a subset of a vector space that is itself a vector space. Subspaces must pass some simple tests:

  • If u and v are in a subspace, then u + v must also be in a subspace.
  • If c is any scalar, and v is in the subspace, then c * v must also be in the subspace.
  • The zero vector must be in any subspace.

Example 1: Lines through the origin

Any line in R^2 passing through the origin is a subspace. It contains the zero vector and is closed under addition and scalar multiplication.

Example 2: Plane

The subspace of R^3 can be any plane through the origin. Again, it has the zero vector and is closed under our operations.

Base and dimensions

A basis for a vector space V is a set of vectors in V that are linearly independent and that span V. This means that you can reach any vector in V by scaling and adding the basis vectors. The dimension of a vector space is the number of vectors in the basis of the vector space. For example, a basis of R^2 typically consists of two vectors, such as (1, 0) and (0, 1), and thus has dimension 2.

(1,0) (0,1)

These vectors form a basis for R^2.

In three dimensions, R^3, a simple basis could be (1, 0, 0), (0, 1, 0), and (0, 0, 1).

Spanning sets

A set of vectors {v1, v2, ..., vn} is said to be spanned in a vector space V if every vector in V can be written as a linear combination of v1, v2, ..., vn. Spanning sets include definitions and concepts for basis.

Linear independence

Vectors are linearly independent if none of them can be written as a linear combination of the others. This independence is a key concept in defining the dimension of a vector space.

Mathematical representation

For vectors v1, v2, ..., vn in a vector space, they are linearly independent if the equation:

a1*v1 + a2*v2 + ... + an*vn = 0

It has only the trivial solution where a1 = a2 = ... = an = 0.

Conclusion

Vector spaces are the cornerstone of linear algebra and are used throughout mathematics and the applied sciences. They provide a formal method for studying linear equations and transformations, providing insight into many real-world problems structured around the structure of space and multi-dimensional quantities.


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Vector Spaces

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