Definitions

# linear algebra

Branch of algebra concerned with methods of solving systems of linear equations; more generally, the mathematics of linear transformations and vector spaces. “Linear” refers to the form of the equations involved—in two dimensions, math.amath.x + math.bmath.y = math.c. Geometrically, this represents a line. If the variables are replaced by vectors, functions, or derivatives, the equation becomes a linear transformation. A system of equations of this type is a system of linear transformations. Because it shows when such a system has a solution and how to find it, linear algebra is essential to the theory of mathematical analysis and differential equations. Its applications extend beyond the physical sciences into, for example, biology and economics.

Generalized version of arithmetic that uses variables to stand for unspecified numbers. Its purpose is to solve algebraic equations or systems of equations. Examples of such solutions are the quadratic formula (for solving a quadratic equation) and Gauss-Jordan elimination (for solving a system of equations in matrix form). In higher mathematics, an “algebra” is a structure consisting of a class of objects and a set of rules (analogous to addition and multiplication) for combining them. Basic and higher algebraic structures share two essential characteristics: (1) calculations involve a finite number of steps and (2) calculations involve abstract symbols (usually letters) representing more general objects (usually numbers). Higher algebra (also known as modern or abstract algebra) includes all of elementary algebra, as well as group theory, theory of rings, field theory, manifolds, and vector spaces.

Symbolic system used for designing logic circuits and networks for digital computers. Its chief utility is in representing the truth value of statements, rather than the numeric quantities handled by ordinary algebra. It lends itself to use in the binary system employed by digital computers, since the only possible truth values, true and false, can be represented by the binary digits 1 and 0. A circuit in computer memory can be open or closed, depending on the value assigned to it, and it is the integrated work of such circuits that give computers their computing ability. The fundamental operations of Boolean logic, often called Boolean operators, are “and,” “or,” and “not”; combinations of these make up 13 other Boolean operators.

B*-algebras are mathematical structures studied in functional analysis.

### General Banach *-algebras

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

1. (x + y)* = x* + y* for all x, y in A.
2. $\left(lambda x\right)^* = bar\left\{lambda\right\}x^*$ for every λ in C and every x in A; here, $bar\left\{lambda\right\}$ denotes the complex conjugate of λ.
3. (xy)* = y* x* for all x, y in A.
4. (x*)* = x for all x in A.

In most natural examples, one also has that the involution is isometric, i.e.

• ||x*|| = ||x||,

### B* algebras

A B*-algebra is a Banach *-algebra in which the involution satisfies the following further property:

• ||x x*|| = ||x||2 for all x in A.

By a theorem of Gelfand and Naimark, given a B* algebra A there exists a Hilbert space H and an isometric *-homomorphism from A into the algebra B(H) of all bounded linear operators on H. Thus every B* algebra is isometrically *-isomorphic to a C*-algebra. Because of this, the term B* algebra is rarely used in current terminology, and has been replaced by the (overloading of) the term 'C* algebra'.