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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

Matrix addition

How to add two matrices together, definition and properties of addition

Vectors and matrices

Matrices, their characteristics, introduction to some special matrices

Multiplication of a matrix by a scalar

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

Matrix multiplication

How to multiply two matrices, definition and properties of multiplication

Identity matrix

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

Linear combinations and linear spaces

Linear spaces

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

Linear combinations

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

Linear span

The linear space generated by a set of vectors

Linear independence

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

Dimension of a linear space

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

Basis of a linear space

A set of linearly independent vectors that span the linear space

Matrix product and linear combinations

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

Standard basis

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

Matrix product and rank

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

Rank of a matrix

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

Matrix inversion lemmas

Formulae for computing how changes in a matrix affect its inverse

Inverse of a matrix

Multivariate generalization of the concept of reciprocal of a number

Linear systems

Equivalent systems of equations

Systems of linear equations having the same set of solutions

Matrices and linear systems

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

Augmented matrix

A compact way to represent systems of linear equations

Elementary row operations

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

Gaussian elimination

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

Row echelon form

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

Gauss-Jordan elimination

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

Reduced row echelon form

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

Elementary column operations

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

Special matrices

Triangular matrix

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

Permutation matrix

A matrix used to perform multiple interchanges of rows and columns

Diagonal matrix

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

Elementary matrix

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


LU decomposition

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

Row equivalence

How elementary row operations generate equivalence classes


Trace of a matrix

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


Determinant of a matrix

A number telling us how the associated linear transformation scales volumes

Sign of a permutation

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

Properties of the determinant

Discover several properties enjoyed by the determinant of a matrix

Determinants of elementary matrices

Determinants of elementary matrices enjoy some special properties

Eigenvalues and eigenvectors

Eigenvalues and eigenvectors

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

Complex vectors and matrices

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.