We have chosen to give it at the end of the section since it deals with differential equations and thus is not purely linear algebra.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

The same definition holds in any unital ring or algebra where a is any invertible element.

In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

His notion of abelian category is now the basic object of study in homological algebra.

With the existence of an alpha channel, it is possible to express compositing image operations, using a compositing algebra.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Explicitly, is an associative algebra homomorphism if

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.

* Any ring of characteristic n is a ( Z / nZ )- algebra in the same way.

* Any ring A is an algebra over its center Z ( A ), or over any subring of its center.

* Any commutative ring R is an algebra over itself, or any subring of R.

Other important Arabic astrologers include Albumasur and Al Khwarizmi, the Persian mathematician, astronomer and astrologer, who is considered the father of algebra and the algorithm.

* Another example of a Boolean algebra that is not complete is the Boolean algebra P ( ω ) of all sets of natural numbers, quotiented out by the ideal Fin of finite subsets.

* The square n-by-n matrices with entries from the field K form a unitary associative algebra over K.

* The complex numbers form a 2-dimensional unitary associative algebra over the real numbers.

* The quaternions form a 4-dimensional unitary associative algebra over the reals ( but not an algebra over the complex numbers, since if complex numbers are treated as a subset of the quaternions, complex numbers and quaternions do not commute ).

* The polynomials with real coefficients form a unitary associative algebra over the reals.

In abstract algebra, a field extension L / K is called algebraic if every element of L is algebraic over K, i. e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i. e. which contain transcendental elements, are called transcendental.

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space.

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.

A Banach *- algebra A is a Banach algebra over the field of complex numbers, together with a map *: A → A called involution which has the following properties:

A C *- algebra, A, is a Banach algebra over the field of complex numbers, together with a map *: A → A.

The algebra M < sub > n </ sub >( C ) of n-by-n matrices over C becomes a C *- algebra if we consider matrices as operators on the Euclidean space, C < sup > n </ sup >, and use the operator norm ||.|| on matrices.

Much of linear algebra may be formulated, and remains correct, for ( left ) modules over division rings instead of vector spaces over fields.

Differences between linear algebra over fields and skew fields occur whenever the order of the factors in a product matters.

This reduction has been accomplished by the general methods of linear algebra, i.e., by the primary decomposition theorem.

solution for the additive coefficients by simple algebra rather than

100 cc and P = 2. 50E6 pascals, so we can solve for temperature by simple algebra:

The algebra A can then be thought of as an R-module by defining

The complexity of this law served as an impetus behind the development of algebra ( Arabic: al-jabr ) by the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī and other medieval Islamic mathematicians.

Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative.

A generalization of Artin's theorem states that whenever three elements in an alternative algebra associate ( i. e. ) the subalgebra generated by those elements is associative.

The close relationship of alternative algebras and composition algebras was given by Guy Roos in 2008: He shows ( page 162 ) the relation for an algebra A with unit element e and an involutive anti-automorphism such that a + a * and aa * are on the line spanned by e for all a in A.

Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra.

Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + ( x · y ) and x ∧ y := x · y.

Is X a Banach space, the space B ( X ) = B ( X, X ) forms a unital Banach algebra ; the multiplication operation is given by the composition of linear maps.

The algebra multiplication and the Banach space norm are required to be related by the following inequality:

Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering and then applying the outcome in the original algebra.

* The set of real ( or complex ) numbers is a Banach algebra with norm given by the absolute value.

* The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.

* Measure algebra: A Banach algebra consisting of all Radon measures on some locally compact group, where the product of two measures is given by convolution.

where is the Gelfand representation of x defined as follows: is the continuous function from Δ ( A ) to C given by The spectrum of in the formula above, is the spectrum as element of the algebra C ( Δ ( A )) of complex continuous functions on the compact space Δ ( A ).