linear algebra

linear algebra

Linear algebra

is the branch of mathematics devoted to the theory of
linear

structure

. The axiomatic treatment of linear structure is
based on the notions of a

linear space

(more commonly known as a

vector space

), and a

linear mapping

. Broadly speaking,
there are two fundamental questions considered by linear algebra:

•

the

solution

of a

linear equation

, and

•

diagonalization

, a.k.a. the

eigenvalue problem

.

From the geometric point of view, “linear” is synonymous with
“

straight

”, and consequently linear algebra can be regarded as the
branch of mathematics dealing with lines and planes, as well as with

transformations

of space that preserve “straightness”, e.g.

rotations

and

reflections

. The two fundamental questions, in geometric
terms, deal with

•

the

intersection

of

hyperplanes

, and

•

the principal axes of an

ellipsoid

.

Linearity is a very basic notion, and consequently linear algebra has
applications in numerous areas of mathematics, science, and
engineering. Diverse disciplines, such as differential equations,

differential geometry

, the theory of relativity, quantum mechanics,
electrical

circuits

, computer graphics, and information theory benefit
from the notions and techniques of linear algebra.

Euclidean geometry

is related to a specialized branch of linear
algebra that deals with linear measurement. Here the relevant notions
are length and angle. A typical question is the determination of
lines

perpendicular

to a given plane. A somewhat less specialized
branch deals with affine structure, where the key notion is that of
area and volume. Here

determinants

play an essential role.

Yet another branch of linear algebra is concerned with computation,
algorithms, and numerical approximation. Important examples of such
techniques include:

Gaussian elimination

, the method of least squares,

LU factorization

,

QR decomposition

,

Gram-Schmidt orthogonalization

,

singular value decomposition

, and a number of iterative algorithms for
the calculation of eigenvalues and eigenvectors.

The following subject outline surveys
key topics in linear algebra.

1.

Linear structure.

(a)

Introduction

:

systems of linear equations

, Gaussian
elimination, matrices,

matrix operations

.

(b)

Foundations

:

fields and vector spaces,

subspace

,

linear
independence

, basis,

ordered basis

,

dimension

,

direct sum

decomposition.

(c)

Linear mappings:

linearity axioms, kernels and images,
injectivity, surjectivity,

bijections

,

compositions

,

inverses

,

matrix representations

,

change of bases

,

conjugation

,

similarity

.

2.

Affine structure.

(a)

Determinants:

characterizing properties,

cofactor
expansion

,

permutations

, Cramer’s rule,

classical adjoint

.

(b)

Geometric aspects:

Euclidean

volume,

orientation

,
equiaffine transformations, determinants as geometric

invariants

of linear transformations.

3.

Diagonalization and Decomposition.

(a)

Basic notions:

eigenvector

,

eigenvalue

,

eigenspace

,

characteristic polynomial

.

(b)

Obstructions:

imaginary

eigenvalues,

nilpotent
transformations

,

classification

of 2-dimensional real
transformations.

(c)

Structure theory:

invariant subspaces

,

Cayley-Hamilton
theorem

,

Jordan canonical form

,

rational canonical form

.

4.

Multi-linearity

.

(a)

Foundations:

vector space dual, bilinearity,

bilinear

transpose

, Gram-Schmidt orthogonalization.

(b)

Bilinearity:

bilinear forms

,

symmetric bilinear forms

,

quadratic forms

,

signature

and Sylvester’s

theorem

,

orthogonal
transformations

,

skew-symmetric bilinear forms

, symplectic
transformations.

(c)

Tensor algebra

:

tensor product

,

contraction

, invariants
of linear transformations,

symmetry operations

.

5.

Euclidean and

Hermitian

structure.

(a)

Foundations:

inner product

axioms, the

adjoint

operation

,

symmetric

transformations,

skew-symmetric

transformations,

self-adjoint

transformations, normal transformations.

(b)

Spectral theorem

:

diagonalization of self-adjoint
transformations,

diagonalization of quadratic forms

.

6.

Computational and numerical methods.

(a)

Linear problems:

LU-factorization, QR decomposition, least
squares,

Householder transformations

.

(b)

Eigenvalue problems:

singular value decomposition, Gauss and
Jacobi-Siedel iterative algorithms.

Title

linear algebra

Canonical name

LinearAlgebra

Date of creation

2013-03-22 12:26:16

Last modified on

2013-03-22 12:26:16

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

10

Author

rmilson (146)

Entry type

Topic

Classification

msc 15-00

Classification

msc 15-01

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