Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Abstract Algebra ↓


Linear Algebra


Linear algebra is a cornerstone of modern mathematics and its applications in science, engineering, and many other fields. It deals with vectors, matrices, vector spaces, and linear transformations. Even though it is a branch that starts with basic concepts, its implications are vast and powerful, making it an important topic in undergraduate mathematics studies, especially within abstract algebra.

Basic concepts

In linear algebra, we start by understanding simple but powerful structures called vectors and matrices. Let's define what these are:

Vectors

A vector is essentially an ordered list of numbers. We can think of it as a point in space, where each number represents a coordinate on an axis. A vector in two-dimensional space can be written as:

V = [x, y]

For example, a vector might be [3, 4]. This represents a point in 2D space that is 3 units along the x-axis and 4 units along the y-axis.

Matrices

A matrix is a rectangular array of numbers, which can be used to represent a collection of vectors or to describe transformations. A matrix with two rows and two columns can be represented as:

A = [[a, b],
     [c, d]]

For example, a matrix might look like this:

A = [[1, 2],
     [3, 4]]

This matrix has two rows and two columns, often called a 2x2 matrix.

Operations on vectors and matrices

There are several important operations we can perform with vectors and matrices, including addition, scalar multiplication, and the important matrix multiplication. To understand these operations, let’s look at them one by one:

Vector addition

Adding two vectors involves adding their corresponding elements. If you have two vectors:

u = [u1, u2]
v = [v1, v2]

Their sum, u + v, is defined as follows:

u + v = [u1 + v1, u2 + v2]

For example, if u = [1, 3] and v = [4, 5], then:

  u + v = [1 + 4, 3 + 5] = [5, 8]

Scalar multiplication

Scalar multiplication involves multiplying each component of a vector by a scalar (a single number). If c is a scalar and v = [v1, v2] is a vector, then scalar multiplication is c * v:

c * v = [c * v1, c * v2]

If the vector v = [2, 3] and the scalar c = 4, the result of the operation will be:

  4 * [2, 3] = [8, 12]

Matrix multiplication

Matrix multiplication is perhaps one of the most complex operations in linear algebra, yet it is fundamental. If A is an m x n matrix and B is an n x p matrix, then their product AB is an m x p matrix. The element in the i-th row and j-th column of AB is calculated as follows:

(AB)_{ij} = Σ (A_{ik} * B_{kj})

Example: Let A and B be:

  A = [[1, 2],
       [3, 4]]

  B = [[5, 6],
       [7, 8]]

The product AB is calculated as follows:

  AB = [[(1*5 + 2*7), (1*6 + 2*8)],
        [(3*5 + 4*7), (3*6 + 4*8)]]

     = [[19, 22],
        [43, 50]]

Vector space

A vector space is a collection of vectors that can be added together and scaled by numbers, known as scalars. Scalars are usually real numbers, but complex numbers or other numerical fields can also be used.

Important properties that characterize a vector space include:

  • Closure under addition: The sum of any two vectors in space is also a vector in space.
  • Closure under scalar multiplication: The product of any scalar with a vector in space produces another vector in the space.
  • Existence of the zero vector: There is a vector in the space that behaves as an identity element for the sum.
  • Existence of additive inverses: For every vector in space, there exists another vector which, when added together, will give the zero vector.

Linear independence, basis and dimension

Now, let's look at some advanced concepts important for understanding vector spaces:

Linear independence

A set of vectors is called linearly independent if no vector in the set can be written as a linear combination of other vectors. For example, vectors v1, v2, and v3 are linearly independent if:

c1 * v1 + c2 * v2 + c3 * v3 = 0

This is applicable only if c1 = c2 = c3 = 0.

Basis

A basis of a vector space is a set of linearly independent vectors that span the entire vector space. With a basis, you can express any vector in space as a linear combination of basis vectors.

Dimensions

The dimension of a vector space is the number of vectors in a basis of that space. For example, any basis of the plane (a two-dimensional vector space) will contain 2 vectors.

Linear transforms

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. If T: V -> W is a linear transformation, and u, v are vectors in V and c is a scalar, then the following must hold:

  • T(u + v) = T(u) + T(v)
  • T(c * u) = c * T(u)

Every linear transformation can be represented by a matrix, and understanding these transformations can greatly simplify complex mathematical problems.

Kernel and image

Let's discuss the kernel and image, which are important concepts for understanding linear transformations:

Kernels

The kernel of a linear transformation T is the set of all vectors v in V such that T(v) = 0 is a subspace of the kernel domain.

Image

The image of a linear transformation T is the set of all vectors that can be written as T(v) for some vector v in V. The image is a subspace of the codomain.

Eigenvectors and eigenvalues

Eigenvectors and eigenvalues are concepts that are used frequently in linear algebra, especially when working with matrix transformations.

Eigenvectors

The eigenvector of a matrix A is a vector v such that when A is multiplied by v, the product is a scalar multiple of v. It can be written as:

A * v = λ * v

where λ is the eigenvalue associated with the eigenvector v.

Eigenvalue

The eigenvalue is a scalar λ corresponding to the eigenvector v, such that the equation A * v = λ * v is valid. To find the eigenvalue, we solve the characteristic equation:

det(A - λI) = 0

where I is the identity matrix of the same dimension as A

Applications of linear algebra

Linear algebra has wide applications in various fields:

  • Computer graphics: For the transformation and representation of 3D objects.
  • Machine learning: Linear models and neural networks.
  • Statistics: Describing large sets of data points through methods such as PCA.
  • Engineering: For control systems, signal processing, etc.
  • Physics: Quantum mechanics, where linear operators play an essential role.

This is just a glimpse of the vast content and depth of linear algebra. This discipline is the basis of modern technological and scientific advancements, highlighting its importance and the need for a thorough understanding.


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