Algebra - vocabulary

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.