Linear algebra

Fields

What an algebraic field is, one of the basic structures of linear algebra. The definition, binary operations, how to build a field and more examples.

Relations

A relation is the mathematical counterpart of the everyday idea of a relationship. Arity, unary, binary and n-ary relations, how to write relations, and plenty of examples.

Matrix

What a matrix is and how it is written. Special types of matrix, operations on matrices, multiplication, row operations, rank, and non-singular and singular matrices.

Rank of a matrix

What the rank of a matrix is and how to work it out by reducing to row echelon form. Definition, properties and worked examples, including a matrix with a parameter.

Inverse matrix

What an inverse matrix is and when it exists. How to work it out by row operations and by using the determinant, with properties and worked examples.

Determinants

What the determinant of a matrix is and how to work it out. The sign of a permutation, the definition of the determinant, the rule of Sarrus and Laplace expansion along a row.

Systems of linear equations

How to solve systems of linear equations using matrices. Matrix form, the Rouché–Capelli theorem and Gaussian elimination, with general and particular solutions.

Vector spaces

What a vector space is and which properties it must satisfy. The definition, the space R^2 as an illustrative example, and the size of vector spaces.