 StatLect

Matrix algebra

This is a short course in matrix algebra, with a focus on concepts that are often used in probability and statistics.

Vectors, matrices and basic algebraic operations

Matrices, their characteristics, introduction to some special matrices

How to multiply a matrix by a scalar, definition and properties of scalar multiplication

How to multiply two matrices, definition and properties of multiplication

It plays in matrix multiplication the same role played by 1 in the multiplication of numbers

Linear combinations and linear spaces

Sets of matrices or vectors that are closed with respect to taking linear combinations

Obtained by multiplying matrices by scalars, and by adding them together

The linear space generated by a set of vectors

One of the central concepts in linear algebra and in the theory of linear systems

The number of elements of any one of the bases of the linear space

A set of linearly independent vectors that span the linear space

Multiplying matrices is equivalent to taking linear combinations of their rows and columns

A basis made up of vectors that have all entries equal to zero except one

Discover some useful facts about the rank of the product of two matrices

The dimension of the linear space spanned by the columns or rows of the matrix

Formulae for computing how changes in a matrix affect its inverse

Multivariate generalization of the concept of reciprocal of a number

Linear systems

Systems of linear equations having the same set of solutions

Systems of linear equations can be written compactly and easily studied with matrices

A compact way to represent systems of linear equations

Elementary operations that allow to transform a linear system into an equivalent system

The main algorithm used to reduce linear systems to row echelon form

Systems of linear equations having this form can be easily solved with the back-substitution algorithm

The standard algorithm used to transform linear systems to reduced row echelon form

Echelon form in which the basic columns are vectors of the standard basis

Operations that allow to transform a linear system arranged horizontally into an equivalent system

Special matrices

A matrix that has all entries below (or above) the main diagonal equal to zero

A matrix used to perform multiple interchanges of rows and columns

A matrix whose off-diagonal entries are all equal to zero

A matrix obtained by performing an elementary operation on an identity matrix

Equivalence

How to write a matrix as a product of a lower and an upper triangular matrix

How elementary row operations generate equivalence classes

Trace

The trace of a matrix is the sum of the entries on its main diagonal

Determinants

A number telling us how the associated linear transformation scales volumes

A concept that pops up in the definition of determinant of a matrix

Discover several properties enjoyed by the determinant of a matrix

Determinants of elementary matrices enjoy some special properties

Eigenvalues and eigenvectors

Linear transformations scale up or down the sides of certain parallelograms but do not change their angles

Basic facts and definitions about matrices whose entries are complex numbers

The book

Most of the learning materials found on this website are now available in a traditional textbook format.

Glossary entries
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