Everything about Linear Algebra totally explained
Linear algebra is the branch of
mathematics concerned with the study of
vectors,
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's 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
The history of modern linear algebra dates back to the early 1840s. In 1843,
William Rowan Hamilton introduced
quaternions, which describe mechanics in three-dimensional space. In 1844,
Hermann Grassmann published his book
Die lineale Ausdehnungslehre (see References).
Arthur Cayley introduced
matrices, one of the most fundamental linear algebraic ideas, in 1857. Despite these early developments,
linear algebra has been developed primarily in the twentieth century. It was the focus of one of the first international mathematical societies, the
Quaternion Society (1899 - 1913), which aimed to study
allied systems of mathematics.
Matrices were poorly-defined before the development of
ring theory within
abstract algebra. With the coming of
special relativity, many practitioners gained appreciation of the subtleties of linear algebra. Furthermore, the routine application of
Cramer's rule to solve
partial differential equations led to the inclusion of linear algebra in standard coursework at universities. E.T. Copson wrote, for instance,
Francis Galton initiated the use of
correlation coefficients in 1888. Often more than one
random variable is in play and may be
cross-correlated. In
statistical analysis of
multivariate random variables the
correlation matrix is a natural tool. Thus, statistical study of such random vectors helped establish matrix usage.
More recent developments followed the formulation of the
vector space concept into an
algebraic structure, and the growth of
functional analysis. One can see a diverse set of applications in the
list of matrices.
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, represented by length, and its direction. Vectors can be used to represent physical entities such as
forces, and they can be added to each other and multiplied with
scalars, thus forming the first example of a
real vector space.
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 can't easily visualize vectors in
n-space, such vectors or
n-tuples are useful in representing data. Since vectors, as
n-tuples, are
ordered lists of
n components, it's possible to summarize and manipulate data efficiently 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,
Spain,
India,
Japan,
Australia), 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's 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 transform 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.
The general method of finding a linear way to look at a problem, expressing this in terms of linear algebra, and solving it, if need be by matrix calculations, is one of the most generally applicable in mathematics.
Some useful theorems
Generalisations 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. In the spectral theory of operators control of infinite-dimensional matrices is gained, by applying
mathematical analysis in a theory that isn't purely algebraic. In all these cases the technical difficulties are much greater.
Further Information
Get more info on 'Linear Algebra'.
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