Algebra - vocabulary

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.