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

noun

Pronunciation:

[dɪsˌtrɪbjʊˈtɪvɪti]

Meaning:

a property connecting addition and multiplication; for all numbers a, b, c it holds that a(b+c) = ac + bc and (a+b)c = ac + bc

Sources:

https://www.oed.com/view/Entry/55790?redirectedFrom=distributivity#eid

https://www.oed.com/view/Entry/55787#eid6517953

https://mathworld.wolfram.com/Distributive.html

noun

Pronunciation:

[dɒt 'prɒdʌkt]

Meaning:

the sum of the products of corresponding coordinates of two real vectors, or of the products of the coordinates of the second of two complex vectors and the complex conjugates of the corresponding coordinates of the first

Source:

https://www.oed.com/view/Entry/56962?redirectedFrom=dot+product#eid1265788880

n.

Pronunciation

ˈīgənˌvalyo͞o

Meaning

One of those special values of a parameter in an equation for which the equation has a solution.

Etymology

translating German eigenwert

www.oed.com

Noun
Prefix: eigen-

Pronunciation
/'ʌɪɡ(ə)nˌvɛktə/ listen

Synonym
Characteristic vector of a linear transformation

Meaning
An eigenvector is a nonzero vector that is changed at most by a scalar factor after the application of a linear transformation. It satisfies the formula Av = λv, where A is a matrix, λ is a scalar (eigenvalue) and v is the eigenvector.

Example
This figure displays the matrix A acting on the vector x. The direction of the vector remains; thus, the x is the eigenvector of A.

Etymology
German eigen (own), from adoptions or partial translations of German compounds in Mathematics and Physics.

Sources
OED
Figure from WIKIPEDIA

Noun
Suffix: -ing

Pronunciation
/ɪm'bɛdiŋ/ listen

Meaning
The action of giving some mathematical structure in another (a subgroup in a group).

Example
The map f is an embedding in the following:

The embedding is denoted by the use of a hooked arrow.

Similar words:
embed (also imbed)                         verb              listen
embedded (also imbedded)          adjective     listen 
embeddable (also imbeddable)    adjective     listen

Etymology
First in the paper 1939 Duke Math. Jrnl. Isometric embedding of flat manifolds in Euclidian space.

Source
OED

noun

Pronunciation:

/fiːld/

Definition:

Field is an algebraic structure defined as 7-tuple of set, binary operations on this set (addition and multiplication), two unary operations (multiplicative inverse) and two nullary operations (0 and 1). With folowing axioms:

• associativity of addition and multiplication
• commutativity of addition and multiplication
• 0 and 1 are additive and multiplicative identity, it means that  a + 0 = a and a· 1 = a for all a from field
• for  additive inverse (-a) holds: a + (−a) = 0
• for multiplicative inverse (a^(-1)) holds: a· a⁻¹ = 1
• distributivity of multiplication over addition: a· (b + c) = (a· b) + (a· c)

source:

https://www.macmillandictionary.com

https://en.wikipedia.org

noun

pronunciation:

[ fēld əv ˈfrakSHəns ]

synonyms:

fraction field, field of quotients, or quotient field

meaning:

 Field of fractions of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain  are equivalence classes (see the construction below) written as  with  and  in  and . The field of fractions of  is sometimes denoted by  or.

examples:

The field of fractions of the ring of integers is the field of rationals, i.e. .

Given a field , the field of fractions of the polynomial ring in one indeterminate  (which is an integral domain), is called the field of rational functions or field of rational fractions and is denoted .

Gaussian, adj.

Elimination, n. (created from a verb eliminate, /ᵻˈlɪmᵻneɪt/, and a suffix -ion)

Algorithm, n.

Pronounciation:

/ˈɡaʊsɪən/, /ᵻˌlɪmᵻˈneɪʃn/, /ˈalɡərɪð(ə)m/

Meaning:

Gaussian: Discovered or formulated by Gauss.

Elimination: The removal of a constant, variable, factor, etc., from a system of equations or a matrix by algebraic manipulation.

Algorithm: A procedure or set of rules used in calculation and problem-solving; (in later use spec.) a precisely defined set of mathematical or logical operations for the performance of a particular task.

Source:

OED

noun

pronunciation:

[ ˈgreytist ˈkämən diˈvīzər dōˈmān ]

meaning:

is an integral domain with the property that any two elements have a greatest common divisor (GCD)

property:

If R is a GCD domain, then the polynomial ring R[X₁,...,X_n] is also a GCD domain.

a set of operations so constituted that the product of any number of these operations is always itself a member of the set. In later use more generally: a set of elements together with an operation for combining any two of them to form a third element which is also in the set, the operation satisfying certain conditions.

Etymology:  French groupe, grouppe small detachment of soldiers (1574), arrangement of two or more figures or objects in a design . 1668), (in music) series of notes forming an ornament, run, etc., or linked by a slur (1703), number of things having some related properties or attributes in common (1726)

https://www.oed.com/

noun

pronunciation:

[ gro͞op ˌhōməˈmôrˌfizəm ]

meaning:

given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that

where the group operation on the left hand side of the equation is that of G and on the right hand side that of H.

From this property, one can deduce that h maps the identity element e_G of G to the identity element e_H of H, 

and it also maps inverses to inverses in the sense that

 The identity matrix is a square diagonal matrix with ones on the diagonal.

/ˈɪm.ɪdʒ/

Let f be a mapping of X to Y. Image of f is set {f(x):x fromX} and is denoted f(X).

Noun
Prefix: in-

Pronunciation
/ɪn'dʒɛkʃən/ listen

Meaning
A class of functions whose each element of codomain is mapped to by at most one element of the domain.

Example

Synonyms
Injective function, one-to-one function

Similar words
Injective adjective /ɪnˈdʒɛktɪv/ listen

Etymology
The term introduced by Nicolas Bourbaki. The prefix in- means in, income.

Source
OED

noun

pronunciation:

[ ˈɪnə ˈprɒdʌkt ]

meaning:

An inner product is a generalization of the dot product. In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar.

More precisely, for a real vector space, an inner product  satisfies the following four properties. Let , , and  be vectors and  be a scalar, then:

1. .

2. .

3. .

4.  and equal if and only if .

example:

For Euclidean space ,  the inner product is given by 

noun

pronunciation:

[ˈintigrəl dōˈmān]

meaning:

is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.

example:

The ring  of all polynomials in one variable with integer coefficients is an integral domain.

property:

In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

adj.

Etymology

classical Latin invertere to turn upside down or inside out, to reverse, to turn over violently, upset, to turn round, to pervert, to reverse (an order), to cause words to convey the opposite sense (e.g. by irony), to change, alter, to paraphrase, to translate < in- in- prefix2 + vertere vert v.1

Pronunciation

Adjective
Prefix:    ir-
Suffix:   -ible

Pronunciation
/ˌɪrɪˈdjuːsɪbəl/ listen

Meaning
A polynomial that cannot be reduced to a simpler form. It cannot be factored into the product of two non-constant polynomials.

Synonym
Prime polynomial

Antonym
Reducible (polynomial) listen
A polynomial that can be converted into a simpler form. It can be written as the product of two polynomials in the field with positive degrees.

Examples

Polynomial (1) is irreducible over the integers but reducible over the reals.
Polynomial (2) is reducible.
Polynomial (3) is irreducible over the reals.

Similar words
to reduce     /rɪˈdjuːs/        verb
reduction     /rɪˈdʌkʃən/    noun
reductive     /rɪˈdʌktɪv/      adjective

Etymology
From post-classical Latin reducibilis (from 13th century in British and continental sources)
From classical Latin redūcere (verb reduce + -ibilis -ible suffix)

Sources
OED

Noun

Pronunciation
/ˈkəːnəl/ listen

Synonym
Null space

Meaning
A kernel is an inverse image of 0, solution of the system of linear equations Ax = 0, where A is a matrix. It is denoted by Ker(A).

Example
The kernel consisting of vectors (x, y, z) satisfies, for instance, this

.

The task is to solve

,

which can be computed by Gaussian elimination.

Etymology
Old English cyrnel, diminutive of corn seed, grain, corn n.1 < Old Germanic kurnilo-. Compare (without umlaut) Middle High German kornel a grain, Middle Dutch cornel coarse meal

Source
OED

Pronunciation:

/ˈlɪniə(r) spæn/

Definition:

Let be given a vector space V over a field K. The span of a set S of vectors is defined as the smallest subspace of V, which contains S.

Matrix, n.

Pronounciation:  

Brit. /ˈmeɪtrɪks/

Plural:

matrices, Brit. /ˈmeɪtrᵻsiːz/

Meaning:

A supporting or enclosing structure.

Source:

OED

Noun

Pronunciation
/'mʌɪnə/ listen

Meaning
Minor of a matrixA is the determinant of a smaller square matrix.  Minor M_ij is obtained from matrix A by removing the i-th row and the j-th column.

Example
Minor of the matrix

can be computed as


Synonym
Subdeterminant

Related terms
Minor expansion of a determinant

Etymology
Anglo-Norman and Old French, Middle French menor, menour, menur smaller, lesser, younger (c1100; also used as noun in plural denoting people under the age of majority (13th cent.))

Source
OED

Noun, adjective
Prefix: mono-
Suffix: -al

Pronunciation
/mɒˈnəʊmɪəl/ listen

Meaning
A monomial is a polynomial which has only one term.

Example
5x³

Antonym
Polynomial

Etymology
The prefix mono- means single + -nomial (in binomial n.); after French monôme monome n. (1691)

Source
OED

For A∈ℂ^(n x m), the Moore-Penrose inverse A^(+)∈ℂ^(m x n) is a matrix, satisfying all of the following conditions:

The Moore-Penrose inverse exists for any A and is unique.

source

 /ˈnɔː.məl ˈsʌbˌɡruːp/

Let G be a group and H a subset of G. H is called normal subgroup if it is a subgroup of G and for everey a element of G holds aH=Ha.

Note: The property of being a subgroup is important, do not forget to validate if searching for normal subgroup.

Noun
Suffix:   -tion

Pronunciation
/ˌpɜːmjʊˈteɪʃən/ listen

Meaning
1) Algebra:

The permutation means the action of rearrangement of the elements for another in a set. More formally: The permutation π in S_n is defined as a bijection from a set S_n onto itself. All permutations of a set with n elements create a symmetric group S_n, where the group operation is function composition. It holds four group axioms for two permutations π and σ in S_n: closure, identity, invertibility, and associativity. The composition of two permutations is not commutative.

2) Combinatorics:

The permutation stands for a number of combinations when the order does not matter following this formula:

Examples

This notation means σ(1) = 2, σ(2) = 5, σ(3) = 4, σ(4) = 3, and σ(5) = 1

This figure depicts the graphical illustration of the notation.

Similar words
to permutate     /'pəːmjʊteɪt/         verb
permutability    /pəˌmjuːtə'bɪlɪti/    noun           condition of being permutable
permutable       /pər'mjutəbəl/       adjective    it is possible to permutate it
permutant         /pə'mjuːtənt/         noun           the result from permutation
permutated       /'pəːmjʊteɪtɪd/      adjective    it has been subjected to permutation
permutating      /'pəːmjʊteɪtɪŋ/      adjective    undergoing permutation

Related terms
Even permutation
Odd permutation

Etymology
From French permutation, Latin permūtātiōn-, permūtātiō, Anglo-Norman permutacioun, Anglo-Norman and Middle French permutacion

Sources
OED
Figure

Noun, (verb - see below)

Pronunciation
/'pɪvət/ listen

Synonym
Pivot element

Meaning
A pivot is an element on the left-hand side of a matrix for solving a system of linear equations using the Gaussian elimination that you want the elements above and below to be zero.

Example
The green 3 is a pivot. The aim is to make yellow numbers into zero.

Similar words
To pivot      /'pɪvət/           verb        to make an element above or below a leading one into a zero
Pivoting      /'pɪvətɪŋ/       noun       a process of  making an element above or below a leading one into zero

Etymology
Middle French, French pivot, hinge (1338; 1174–78 in figurative use in the name of a dance), vertical main root of a fruit tree (1651), officer around whom troops wheel (1752)

Source:
OED

n. and adj.

Pronunciation

Meaning

Originally: an expression consisting of many terms, a multinomial. Now: spec. a sum of one or more terms each consisting of a constant multiplied by one or more variables raised to a positive (or non-negative) integral power (e.g. x4 − 3x2y + 7).

Etymology

Origin: Formed within English, by compounding. Etymons: poly-,  -nomial 

OED

noun

Pronunciation:

[prʌɪm]

Meaning:

a positive integer greater than 1 which can be divided without a remainder only by numbers 1 and itself

Source:

https://www.oed.com/view/Entry/266923#eid71284393

noun

Pronunciation:

/prəˈdʒekʃ(ə)n/

Definition:

The linear operation, which two-times implicated gives the same result as identical operation is called projection.

Sorce:

https://www.macmillandictionary.com

Noun

Pronunciation
/ræŋk/ listen

Meaning
A rank is a property of a matrix that tells the dimension of the vector space generated by its columns. This number is the same as the maximal number of linearly independent columns of the matrix. The rank is usually denoted by rank(A), where A is a matrix.

Example
The matrix that has a rank 1 because any pair of columns is linearly dependent.

Etymology
Anglo-Norman and Old French, Middle French renc, ranc, renke, rang with the meaning line (of soldiers), row (of people)

Sources
OED

noun

Pronunciation:

/rɪŋ/

Definition:

Ring is an algebraic structure defined as 5-tuple of set, binary operations on this set (addition and multiplication), one unary operation (addition inverse) and one nullary operation (0). And the following axioms hold.

• addition is associative and commutative
• multiplication is associative
• 0 is the additive identity
• 1 is the multiplicative identity
• a⋅ (b + c) = (a· b) + (a· c) for all a, b, c in R   
• (b + c) · a = (b· a) + (c· a) for all a, b, c in R   

source:

https://dictionary.cambridge.org

noun

Pronunciation:

[ru:t]

Meaning:

a number z such that the value of the polynomial at z equals 0

Source:

https://mathworld.wolfram.com/PolynomialRoots.html

Scalar, adj. and n.

Pronounciation:

/ˈskeɪlə/, /ˈskeɪlɑː/

Meaning:

Noun: In quaternions, a real number. More widely, a quantity having magnitude but no direction, and representable by a single real number.

Adjective: Of the nature of a scalar.

Source:

OED

A singular matrix is a matrix which is not invertible.

The singular values of a matrix A∈ ℂ^(n x m) are the square roots of eigenvalues of A*A.

source 

A singular value decomposition of a complex mxn matrix A is three matrices U, V, ∑, where U is an mxm complex unitary matrix, V is an nxn complex unitary matrix, and ∑ is an mxn diagonal matrix with non-negative real values on the diagonal, such that A=U∑V*.

source 

 /ˈsʌbˌɡruːp/

Let (G, *, ',e) be a group. Let H be a subset of G. We say, that H is a subgroup of G if e is element of H, and for every a,b elements of H are a*b and a' elements of H.

For pronunciation: https://dictionary.cambridge.org/dictionary/english/subgroup 

Noun
Prefix: sur-

Pronunciation
/səːˈdʒɛkʃən/ listen

Meaning
A class of functions whose each element of codomain is mapped to by at least one element of the domain, i.e. the image and the codomain of the function are equal.

Example

Synonyms
Surjective function, onto mapping, onto

Similar words
Surjective adjective /səː'dʒɛktɪv/ listen

Etymology
The term introduced by Nicolas Bourbaki. The prefix sur- means over, above.

Source
OED

/treɪs/

Let A be a square matrix. The trace(A) is defined as sum of all elements on the main diagonal.

Vector, n.

Pronounciation:

/ˈvɛktə/

Meaning:

1) An ordered set of two or more numbers (interpretable as the co-ordinates of a point); a matrix with one row or one column; also, any element of a vector space.

2) A quantity having direction as well as magnitude, denoted by a line drawn from its original to its final position.

Source:

OED

n.

Pronunciation

/ˈvɛktə speɪs/

Meaning

a group whose elements can be combined with each other and with the elements of a scalar field in the way that vectors can, addition within the group being commutative and associative and multiplication by a scalar being distributive and associative.

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