Linear algebra is a branch of
mathematics concerned with the study of
vector,
vector
spaces (also called
linear spaces),
linear maps (also called
linear
transformations), and
systems of linear equations.
Vector spaces are a central theme in modern
mathematics; thus, linear
algebra is widely used in both
abstract algebra and
functional analysis. Linear algebra also
has a concrete representation in
analytic geometry and it is generalized in
operator theory. It has extensive
applications in the
natural sciences
and the
social sciences, since
nonlinear models can often be approximated by linear ones.
History
Many of the basic tools of linear algebra, particularly those
concerned with the solution of systems of linear equations, date to
antiquity--- see e.g. the history of
Gaussian elimination--- although many
objects were not isolated and considered in their own right until
the 1600s and 1700s (see the history of
determinants). The
method of least squares, first used
by
Gauss in the 1790s, is an early and
significant application of the ideas of linear algebra.
The subject began to take its modern form in the mid-19th century,
which saw many notions and methods of previous centuries abstracted
and generalized as the beginnings of
abstract algebra.
Matrices and
tensors were introduced as abstract mathematical
objects and well studied by the turn of the 20th century. The use
of these objects in
special
relativity,
statistics, and
quantum mechanics did much to spread the
subject beyond pure mathematics.
Remarkably, the 2 × 2
complex
matrices were studied before 2 × 2
real
matrices. The early interest was expressed in terms of
biquaternions and
Pauli algebra. Investigation of the
2 × 2 real matrices revealed the
less common
split-complex
numbers and
dual numbers which are
at variance with the Euclidean nature of the ordinary complex
number plane.
Elementary introduction
Linear algebra had its beginnings in the study of vectors in
Cartesian 2-space and
3-space. A vector, here, is a directed
line
segment, characterized by both its magnitude (also called
length or norm) and its direction. The zero vector is an exception;
it has zero magnitude and no direction. Vectors can be used to
represent physical entities such as
forces,
and they can be added to each other and multiplied by
scalar, thus forming the first example
of a
real vector
space, where a distinction is made between "scalars", in this
case real numbers, and "vectors".
Modern linear algebra has been extended to consider spaces of
arbitrary or infinite dimension. A vector space of dimension
n is called an
n-space. Most of the useful
results from 2- and 3-space can be extended to these higher
dimensional spaces. Although people cannot easily visualize vectors
in
n-space, such vectors or
n-tuples are useful in representing data.
Since vectors, as
n-tuples, consist of n ordered
components, data can be efficiently summarized and manipulated in
this framework. For example, in
economics,
one can create and use, say, 8-dimensional vectors or 8-tuples to
represent the
gross national
product of 8 countries.
One can decide to display the GNP of 8
countries for a particular year, where the countries' order is
specified, for example, (United States, United
Kingdom, France, Germany, Italy, Japan, Switzerland and Belgium), by using a
vector (v_{1}, v_{2}, v_{3}, v_{4},
v_{5}, v_{6}, v_{7}, v_{8}) where
each country's GNP is in its respective position.
A vector space (or linear space), as a purely abstract concept
about which
theorems are proved, is part of
abstract algebra, and is well integrated into this discipline.Some
striking examples of this are the
group of invertible linear maps or
matrices, and the
ring of linear maps of a vector space.
Linear algebra also plays an important part in analysis, notably,
in the description of higher order derivatives in vector analysis
and the study of
tensor products and
alternating maps.
In this abstract setting, the scalars with which an element of a
vector space can be multiplied need not be numbers. The only
requirement is that the scalars form a mathematical structure,
called a
field. In applications,
this field is usually the field of
real
numbers or the field of
complex
numbers.
Linear maps take elements
from a linear space to another (or to itself), in a manner that is
compatible with the addition and scalar multiplication given on the
vector space(s).The set of all such transformations is itself a
vector space.If a
basis for a
vector space is fixed, every linear transformation can be
represented by a table of numbers called a
matrix.The detailed study of the
properties of and
algorithms acting on
matrices, including
determinants and
eigenvectors, is considered to be part
of linear algebra.
One can say quite simply that the
linear
problems of
mathematics—those that
exhibit
linearity in their behavior—are
those most likely to be solved. For example,
differential calculus does a great
deal with linear approximation to functions. The difference from
nonlinear problems is very important in
practice.
Some useful theorems
For more information regarding the invertibility of a matrix,
consult the
invertible
matrix article.
Generalizations and related topics
Since linear algebra is a successful theory, its methods have been
developed in other parts of mathematics. In
module theory one replaces the
field of scalars by a ring. In
multilinear algebra one considers
multivariable linear transformations, that is, mappings which are
linear in each of a number of different variables. This line of
inquiry naturally leads to the idea of the
tensor product.
Functional analysis mixes the methods of
linear algebra with those of
mathematical analysis.
See also
Notes
References
Textbooks
History
- Fearnley-Sander, Desmond, "Hermann Grassmann and the Creation
of Linear Algebra" ( via JSTOR), American Mathematical Monthly
86 (1979), pp. 809–817.
- Grassmann, Hermann, Die lineale Ausdehnungslehre ein neuer
Zweig der Mathematik: dargestellt und durch Anwendungen auf die
übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik,
die Lehre vom Magnetismus und die Krystallonomie erläutert, O.
Wigand, Leipzig, 1844.
Further reading
- Introductory textbooks
- Advanced textbooks
- Study guides and outlines
External links
Online books