# Rank

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## rank

^{1}

**1.**any of the eight horizontal rows of squares on a chessboard

**2.**

*Music*a set of organ pipes controlled by the same stop

**3.**

*Maths*(of a matrix) the largest number of linearly independent rows or columns; the number of rows (or columns) of the nonzero determinant of greatest order that can be extracted from the matrix

## rank

^{2}

*Botany*showing vigorous and profuse growth

## Rank

**1.**

**J**(

**oseph**)

**Arthur**, 1st Baron. 1888--1972, British industrialist and film executive, whose companies dominated the British film industry in the 1940s and 1950s

**2.**

**Otto**. 1884--1939, Austrian psychoanalyst, noted for his theory that the trauma of birth may be reflected in certain forms of mental illness

## rank

a position in a SOCIAL STATUS hierarchy The familiar military usage reflects the wider use, which predates the language of social class.*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Rank

in mathematics. The rank of a matrix is the order of a nonsingular minor of maximum order. It is also equal to the largest number of linearly independent rows or columns of the matrix. The rank remains unchanged under elementary transformations of the matrix—that is, when rows or columns are interchanged, when a row or column is multiplied by a nonzero number, and when rows or columns are added. A system of linear equations has a solution if and only if the rank of the matrix formed from the coefficients of the unknowns is equal to the rank of the augmented matrix, that is, the matrix formed by the addition of a column containing the constant terms to the coefficient matrix. This solution is unique if the rank is equal to the number of unknowns.

## Rank

(Russian, *chin*), the service position held by military personnel and civil servants, to which apply specified official rights and obligations. In prerevolutionary Russia, ranks were conferred according to the Table of Ranks, established by Peter I the Great. The concession of estate rights and privileges was connected with the attainment of a specific rank.

All civil and military ranks were abolished by the decrees of Soviet power of Nov. 10 (23) and Dec. 16 (29), 1917. Class ranks (*chiny klassnye*) have been established in the USSR for workers in the Procurator’s Office.

## Rank

a military formation in which servicemen are ranged side by side in a line. In a two-rank formation, the servicemen of one rank take a position one step behind the men of another rank; the front rank is called the first rank and the rear rank, the second. Both formations may be either close or open. In close formation the intervals between the men within a rank are equal to the width of a hand; in open formation the interval is one step or a distance ordered by the commander.

## rank

[raŋk]*n*-dimensional space is of rank

*r*if it has

*n*

^{ r }components.

*G*is the number of elements in the basis of the quotient group of

*G*over the subgroup consisting of all elements of

*G*having finite period.

*P*is the largest number

*n*for which there exists a sequence

*P*

_{0}=

*P*,

*P*

_{1},

*P*

_{2}, … ,

*P*

_{n }of prime ideals such that

*P*

_{i }is a subset of

*P*

_{i-1}.