Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Algebra ↓


Linear Algebra


Linear algebra is a branch of mathematics that deals with vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. It forms the foundation of many scientific disciplines and is used in a variety of fields such as physics, computer science, engineering, and others.

Introduction to vector

A vector is an object that has both magnitude and direction. You can think of it as an arrow pointing from one point in space to another. Vectors are usually represented by symbols such as a, b, or v.

Vectors in 2D

In two dimensions, a vector can be represented as an ordered pair of numbers, which are its coordinates. For example, the vector v = (3, 4) has components 3 in the x-direction and 4 in the y-direction.

v = (3, 4)
(3, 4)

Vectors in 3D

In three dimensions, a vector is represented as a triplet of numbers: v = (x, y, z). The third component adds depth or height.

v = (3, 4, 5)

Vector operations

Several basic operations can be performed on vectors:

Vector addition

To add two vectors, you just have to add their corresponding components. If a = (a1, a2) and b = (b1, b2), then a + b = (a1 + b1, a2 + b2).

a = (3, 4) b = (1, 2) a + b = (3 + 1, 4 + 2) = (4, 6)
(1, 2) (4, 6)

Scalar multiplication

Scalar multiplication involves scaling a vector by a number called a scalar. For a vector a = (a1, a2) and a scalar c, the product c * a = (c*a1, c*a2).

c = 2 a = (3, 4) c * a = (2 * 3, 2 * 4) = (6, 8)
(6, 8)

Vector space

A vector space is a collection of vectors that are closed under vector addition and scalar multiplication. This means that if you take two vectors in a vector space and add them, the result is also in the vector space. Similarly, if you take a vector in the space and multiply it by a scalar, the result is still in the vector space.

Base and dimensions

A set of vectors forms a basis for a vector space if every vector in the space can be written as a linear combination of the basis vectors. The number of vectors in the basis is called the dimension of the vector space.

For example, in two-dimensional space, the standard basis is {(1, 0), (0, 1)}, which has dimension 2.

Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used to represent linear transformations and systems of linear equations.

Matrix representation

The matrix is usually denoted by a capital letter like A or B and for a 2x2 matrix it looks like this:

A = | 1 2 | | 3 4 |

Matrix operations

Like vectors, we can perform operations on matrices too such as addition, multiplication and finding the inverse.

Matrix addition

Matrices can be added together if they have the same dimensions. To add two matrices, add their corresponding elements.

A = | 1 2 | | 3 4 | B = | 5 6 | | 7 8 | A + B = | 1+5 2+6 | | 3+7 4+8 | A + B = | 6 8 | | 10 12 |

Matrix multiplication

Matrix multiplication is more complex than addition and involves the dot product of rows and columns. If A is an mxn matrix and B is an nxp matrix, then the result C = AB is an mxp matrix.

A = | 1 2 | | 3 4 | B = | 5 6 | | 7 8 | AB = | (1*5 + 2*7) (1*6 + 2*8) | | (3*5 + 4*7) (3*6 + 4*8) | AB = | 19 22 | | 43 50 |

Linear transforms

A linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication. If T is a linear transformation, then for vectors u and v and a scalar c, the following conditions hold: T(u + v) = T(u) + T(v) and T(c * u) = c * T(u).

Determinants and inverses

Determinants

The determinant is a special number that can be calculated from a square matrix. It provides important information about the matrix, such as whether or not it is invertible and the amount of transformation described by the matrix.

A = | ab | | cd | det(A) = ad - bc

Inverse

The inverse of a matrix A is another matrix, denoted by A^-1, so that when multiplied together, they yield the identity matrix. Not all matrices have inverses, and a matrix must be square and have a non-zero determinant to be invertible.

A = | 1 2 | | 3 4 | A^-1 = (1/det(A)) * | d -b | | -ca | A^-1 = (1/(1*4 - 2*3)) * | 4 -2 | | -3 1 | A^-1 = | -2 1 | | 1.5 -0.5 |

Eigenvalues and eigenvectors

Eigenvalues and eigenvectors are important concepts in linear algebra and have applications in many areas of science and engineering. An eigenvector of a square matrix A is a non-zero vector v such that when A is multiplied by v, the result is a scalar multiple of v. This scalar is called the eigenvalue.

Av = λv where: A = matrix v = eigenvector λ = eigenvalue

Applications of linear algebra

Linear algebra has many applications in various fields:

  • Computer Graphics: Linear algebra is widely used in computer graphics to represent and manipulate images and 3D models. Transformations such as scaling, rotation, and translation are easily performed using matrices.
  • Engineering: Engineers use linear algebra to solve systems of linear equations that describe physical phenomena such as electrical circuits and mechanical structures.
  • Physics: Linear algebra is used in physics to mathematically model the physical world. Linear algebra is widely used in topics such as quantum mechanics and relativity.
  • Economics: Economists use linear algebra to analyze economic models and optimize production and distribution processes.

Conclusion

Linear algebra is a fundamental area of mathematics that underlies many modern scientific and engineering disciplines. Understanding basic concepts such as vectors, matrices, linear transformations, and the operations that can be performed on them provides a powerful toolkit for solving complex real-world problems.


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