Algebra - vocabulary

1. the abstract study of number systems and operations within them, including such advanced topics as groups, rings, invariant theory, and cohomology. = abstract algebra

2. a particular type of algebraic structure. Formally, an algebra is a vector space V over a field F with a multiplication, which is distributive, and for every f from F and every x,y from V

Etymology:

from the title of a work written around 825 by the Arabic mathematician known as al-Khowarizmi, entitled al-jebr w' al-muqabalah. In Arabic, al- is the definite article "the." The first noun
in the title is jebr"reunion of broken parts," from the verb jabara"to reunite, to consolidate." The second noun is from the verb qabala, with meanings that include "to place in front of, to balance, to oppose,
to set equal." Together the two nouns describe some of the manipulations so common in algebra: combining like terms, transposing a term to the opposite side of an equation, setting two quantities equal, etc.

[Schwartzman, S. (2012). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms used in English. Washington: The Mathematical Association of America.]

noun

Pronunciation:

/ɐnˈaɪ.ə.leɪtə/

Definition:

Let M be a left modul over a ring R and let S be a subset of M. The annihilator of S is the set of all r from R such that sr is equal to zero (for all s from S).

Source:

www.spellchecker.net

noun

Pronunciation:

[ə,səʊsɪə'tɪvɪti]

Meaning:

a property of a binary operation; if it holds, then with more than one of the same operator, the order of operations does not matter

Example:

if + is an associative operator, then for every three elements a, b, c it holds that a + (b + c) = (a + b) + c

Source:

https://simple.wikipedia.org/wiki/Associativity

Noun
Plural: bases

Pronunciation
/'beɪsɪs/ listen

Meaning
A basis is a set of vectors that generates all elements of the vector space and the vectors in the set are linearly independent.
More formally: Let V be a vector space over a fieldF. A subset M which generates the vector space V and which is a linearly independent subset is called a basis of vector space V.
The subset M satisfies the linear independence property and the spanning property.

Examples
Canonical basis {e₁, e₂, ... e_n} in F ⁿ

The figure shows the basis vectors i,j, k, and the vector a is a linear combination of them.

Etymology
From Latin basis and Greek βάσις

Sources
OED
Figure from WIKIPEDIA

Noun
Prefix: bi-

Pronunciation
/bʌɪ'dʒɛkʃ(ə)n/ listen

Meaning
A class of functions whose each element of codomain is mapped to by exactly one element of the domain, i.e. the function is both injective and surjective.

Example

Synonyms
Bijective function, one-to-one correspondence, invertible function

Similar words
Bijective adjective /bʌɪ'dʒɛktɪv/ listen

Etymology
The term introduced by Nicolas Bourbaki. The prefix bi- means two, twice.

Source
OED

Noun   
Suffix: -ity

Pronunciation
/ˌkɒmjuːtə'tɪvɪti/ listen

Meaning
The commutativity is a property of a binary operation. If the property holds, the change of the order of the operands does not change the result.

Example
For addition:               a + b = b + a
For multiplication:      ab = ba

Synonym
Commutative property

Antonym
Noncommutativity

Similar words
To commute      verb          /kɒ'mjuːt/           listen
Commutable     adjective   /kəˈmjuːtəb(ə)l/  listen
Commutative    adjective   /kɒˈmjuːtətɪv/     listen
Commuted       adjective   /kɒ'mjuːtɪd/        listen

Etymology
From medieval Latin commūtātīvus (French (14th cent.) commutatif , -ive ), < Latin commūtāt- participial stem of commūtāre to

Source
OED

 /kənˈstrʌk.ʃən/

A finite sequence of steps. The steps are defined by author of the task, classical example is Eucleidian construction by straightedge and compass.

Adjective
Prefix: co-

Pronunciation
/ˈkoʊˌpraɪm/ listen

Meaning
Coprime integers are two or more numbers that have no common factor except unity. In other words, there is no whole number that divides them both without any remainder. This is equivalent to their greatest common divisor being 1.

Examples
The numerator and denominator of a reduced fraction are coprime, for instance, 7/8.
Integers 15 and 8 are coprime. Integers 8 and 6 are not coprime.

Note

• A set of integers can also be called coprime if its elements share no common positive factor except 1.
• A stronger condition on a set of integers is pairwise coprime. It means that numbers n₁ and n₂ are coprime for every pair (n₁, n₂) of different integers in the set. The set {2, 5, 8} is coprime but it is not pairwise coprime since 2 and 8 are not coprime.

Synonyms
Relatively prime, mutually prime

Antonym
Not coprime

Etymology
Derived from prime, that is partly from French, partly from Latin (French prime; Latin prīmus)
The prefix co- is from Latin. It means together, mutually, jointly, etc.

Source
OED

Noun

Pronunciation
/dɪˈtəːmɪnənt/ listen

Meaning
The determinant is a scalar value computed from the square matrix. It is denoted by det(A) or |A|, where A is a matrix. It is also used for determining the areas or volumes. For instance, the area of a parallelogram can be computed as the absolute value of the determinant of the matrix formed by the vectors representing the sides of the parallelogram.

Examples
The formula for the determinant of a 2 x 2 matrix:

Sarrus' scheme for the determinant of a 3 x 3 matrix:

Leibnitz formula for the determinant of an n x n matrix:


Etymology
Latin dētermināntem, present participle of dētermināre (to determine), French déterminant used in the paper Trevoux 1752

Sources
OED
Formulae: from WIKIPEDIA

Dimension, n.

Pronounciation:

/dɪˈmɛnʃən/

Meaning:

1) Geometry. A mode of linear measurement, magnitude, or extension, in a particular direction; usually as co-existing with similar measurements or extensions in other directions.

2) Algebra. Since the product of two, or of three, quantities, each denoting a length (i.e. a magnitude of one dimension), represents an area or a volume (i.e. a magnitude of two, or of three, dimensions), such products themselves are said to be of so many dimensions; and generally, the number of dimensions of a product is the number of the (unknown or variable) quantities contained in it as factors (known or constant quantities being reckoned of no dimensions); any power of a quantity being of the dimensions denoted by its index.

Source:

OED