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

* The 2 × 2 real matrices form an associative algebra useful in plane mapping.

* The set of all real or complex n-by-n matrices becomes a unital Banach algebra if we equip it with a sub-multiplicative matrix norm.

When the Banach algebra A is the algebra L ( X ) of bounded linear operators on a complex Banach space X ( e. g., the algebra of square matrices ), the notion of the spectrum in A coincides with the usual one in the operator theory.

The geometric interpretation of curl as rotation corresponds to identifying bivectors ( 2-vectors ) in 3 dimensions with the special orthogonal Lie algebra so ( 3 ) of infinitesimal rotations ( in coordinates, skew-symmetric 3 × 3 matrices ), while representing rotations by vectors corresponds to identifying 1-vectors ( equivalently, 2-vectors ) and so ( 3 ), these all being 3-dimensional spaces.

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.

In linear algebra, if two endomorphisms of a space are represented by commuting matrices with respect to one basis, then they are so represented with respect to every basis.

If the 3 generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional ( grassman -) Jordan algebra, which has the symmetry group of one of the exceptional Lie groups ( F < sub > 4 </ sub >, E < sub > 6 </ sub >, E < sub > 7 </ sub > or E < sub > 8 </ sub >) depending on the details.

In particular, the associative algebra of n × n matrices over a field F gives rise to the general linear Lie algebra The associative algebra A is called an enveloping algebra of the Lie algebra L ( A ).

* The subspace of the general linear Lie algebra consisting of matrices of trace zero is a subalgebra, the special linear Lie algebra, denoted

The Lie algebra consists of those matrices X for which

This is a matrix Lie group, and its Lie algebra consists of all n × n matrices with real entries and trace 0.

* The real vector space of all n × n skew-hermitian matrices is closed under the commutator and forms a real Lie algebra denoted.

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.

* 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.

* 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.

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.

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.

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

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

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

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

This loop of units in an alternative ring or algebra is analogous to the group of units in an associative ring or algebra.

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.

Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa.

His mathematical specialties were noncommutative ring theory and computational algebra and its applications, including automated theorem proving in geometry.

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

# Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket = XY − YX, because the Lie bracket of any two derivations is a derivation.

Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism.

* The algebra of matrices over a ring, thought of as a category as described in the article Additive category.

In linear algebra, it can be thought of as a tensor, and is written.

Let be a Lie group and be its Lie algebra ( thought of as the tangent space to the identity element of ).

Just as a Lie groupoid can be thought of as a " Lie group with many objects ", a Lie algebroid is like a " Lie algebra with many objects ".

More formally, a skein relation can be thought of as defining the kernel of a quotient map from the planar algebra of tangles.

The notion of " algebras for a monad " generalizes classical notions from universal algebra, and in this sense, monads can be thought of as " theories ".

In linear algebra terms, the use of a block matrix corresponds to having a linear mapping thought of in terms of corresponding ' bunches ' of basis vectors.

One of the simplest non-trivial examples is a linear complex structure, which is a representation of the complex numbers C, thought of as associative algebra over the real numbers R. This algebra is realized concretely as which corresponds to.

In fact, since any rep of G can be thought of as a module for C and vice versa, we can look at the center of C. This is analogous to looking at the center of the universal enveloping algebra of a Lie algebra.

* Matrix ring, thought of as an algebra over a field or a commutative ring

Leibniz conceived of a characteristica universalis ( also see mathesis universalis ), an " algebra " capable of expressing all conceptual thought.

More formally, a linear complex structure on a real vector space is an algebra representation of the complex numbers C, thought of as an associative algebra over the real numbers.

The most important examples are those with entries in a commutative superalgebra ( such as a Grassmann algebra ) or an ordinary field ( thought of as a purely even commutative superalgebra ).