coefficient

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a coefficient is a multiplicative factor in some term of a
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, a
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
, or any expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are variables, they are often called
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whol ...

s. For example, $2x^2-x+3$, has the real coefficients 2, -1, and 3 respectively, and $ax^2+bx+c$, has coefficient parameters a, b, and c respectively- assuming x is the variable of the equation. The constant coefficient is the coefficient not attached to variables in an expression. For example, the constant coefficients of the expressions above are the real coefficient 3 and the parameter represented by ''c''. Similarly, the coefficient attached to the highest multiplicity of the variable in a polynomial is referred to as the leading coefficient. For example in the expressions above, the leading coefficients are 2 and the parameter represented by ''a''. The
binomial coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s occur in the expanded form of $\left(x+y\right)^n$, and are tabulated in
Pascal's triangle In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
.

# Terminology and Definition

In mathematics, a coefficient is a multiplicative factor in some term of a
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, a
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
, or any expression; For example, in :$7x^2-3xy+1.5+y,$ the first two terms have the coefficients 7 and −3, respectively. The third term 1.5 is a constant coefficient. The final term does not have any explicitly-written coefficient factor that would not change the term; the coefficient is thus taken to be 1 (since variables without number have a coefficient of 1). In many scenarios, coefficients are numbers (as is the case for each term of the above example), although they could be parameters of the problem—or any expression in these parameters. In such a case, one must clearly distinguish between symbols representing variables and symbols representing parameters. Following
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

, the variables are often denoted by , , ..., and the parameters by , , , ..., but this is not always the case. For example, if is considered a parameter in the above expression, then the coefficient of would be , and the constant coefficient (always with respect to ) would be . When one writes :$ax^2+bx+c,$ it is generally assumed that is the only variable, and that , and are parameters; thus the constant coefficient is in this case. Similarly, any
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

in one variable can be written as :$a_k x^k + \dotsb + a_1 x^1 + a_0$ for some positive integer $k$, where $a_k, \dotsc, a_1, a_0$ are coefficients; to allow this kind of expression in all cases, one must allow introducing terms with 0 as coefficient. For the largest $i$ with $a_i \ne 0$ (if any), $a_i$ is called the leading coefficient of the polynomial. For example, the leading coefficient of the polynomial :$\, 4x^5 + x^3 + 2x^2$ is 4. Some specific coefficients that occur frequently in mathematics have dedicated names. For example, the
binomial coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s occur in the expanded form of $\left(x+y\right)^n$, and are tabulated in
Pascal's triangle In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
.

# Linear algebra

In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
, a
system of linear equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is associated with a
coefficient matrixIn linear algebra, a coefficient matrix is a matrix consisting of the coefficient In mathematics, a coefficient is a multiplicative factor in some Summand, term of a polynomial, a series (mathematics), series, or any expression (mathematics), exp ...
, which is used in
Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical s ...
to find a solution to the system. The leading entry (sometimes ''leading coefficient'') of a row in a matrix is the first nonzero entry in that row. So, for example, given the matrix described as follows: :$\begin 1 & 2 & 0 & 6\\ 0 & 2 & 9 & 4\\ 0 & 0 & 0 & 4\\ 0 & 0 & 0 & 0 \end,$ the leading coefficient of the first row is 1; that of the second row is 2; that of the third row is 4, while the last row does not have a leading coefficient. Though coefficients are frequently viewed as constants in elementary algebra, they can also be viewed as variables as the context broadens. For example, the
coordinates In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

$\left(x_1, x_2, \dotsc, x_n\right)$ of a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
$v$ in a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
with
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
$\lbrace e_1, e_2, \dotsc, e_n \rbrace$, are the coefficients of the basis vectors in the expression :$v = x_1 e_1 + x_2 e_2 + \dotsb + x_n e_n .$

*
Correlation coefficient A correlation coefficient is a numerical measure of some type of correlation and dependence, correlation, meaning a statistical relationship between two variable (mathematics), variables. The variables may be two column (database), columns of a give ...

*
Degree of a polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
*
Monic polynomial In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...