Algebra - vocabulary

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D

Distributivity

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

Dot product

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

E

eigenvalue

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

Eigenvector

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

Embedding (also imbedding)

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

F

field

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

Field of fractions

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 $R$ are equivalence classes (see the construction below) written as ${\frac {a}{b}}$ with $a$ and $b$ in $R$ and $b\neq 0$ . The field of fractions of $R$ is sometimes denoted by $\operatorname {Frac} (R)$ or $\operatorname {Quot} (R)$ .

examples:

The field of fractions of the ring of integers is the field of rationals, i.e. $\mathbb {Q} =\operatorname {Frac} (\mathbb {Z} )$

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

G

Gaussian elimination algorithm

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

Greatest Common Divisor (GCD) domain

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.

Group /ɡruːp/

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/

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