Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression. In the latter case, the variables appearing in the coefficients are often called parameters, and must be clearly distinguished from the other variables.

For example, in

$7x^{2}-3xy+1.5+y,$ the first two terms respectively have the coefficients 7 and −3. The third term 1.5 is a constant coefficient. The final term does not have any explicitly written coefficient, but is considered to have coefficient 1, since multiplying by that factor would not change the term.

Often coefficients are numbers as in this 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, the variables are often denoted by x, y, ..., and the parameters by a, b, c, ..., but it is not always the case. For example, if y is considered as a parameter in the above expression, the coefficient of x is −3y, and the constant coefficient is 1.5 + y.

When one writes

$ax^{2}+bx+c,$ it is generally supposed that x is the only variable and that a, b and c are parameters; thus the constant coefficient is c in this case.

Similarly, any polynomial in one variable x 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 $a_{i}\neq 0$ (if any), $a_{i}$ is called the leading coefficient of the polynomial. So 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 received a name. This is the case of the binomial coefficients, the coefficients which occur in the expanded form of $(x+y)^{n}$ , and are tabulated in Pascal's triangle.

Linear algebra

In linear algebra, the leading coefficient (also leading entry) of a row in a matrix is the first nonzero entry in that row. So, for example, given

$M={\begin{pmatrix}1&2&0&6\\0&2&9&4\\0&0&0&4\\0&0&0&0\end{pmatrix}}.$

The leading coefficient of the first row is 1; 2 is the leading coefficient of the second row; 4 is the leading coefficient of the third row, and the last row does not have a leading coefficient.

Though coefficients are frequently viewed as constants in elementary algebra, they can be variables more generally. For example, the coordinates $(x_{1},x_{2},\dotsc ,x_{n})$  of a vector $v$  in a vector space with basis $\lbrace e_{1},e_{2},\dotsc ,e_{n}\rbrace$ , are the coefficients of the basis vectors in the expression

$v=x_{1}e_{1}+x_{2}e_{2}+\dotsb +x_{n}e_{n}.$