Coefficient

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For other senses of this word, see coefficient (disambiguation).

In mathematics, a coefficient is a constant multiplicative factor of a certain object. The object can be such things as a variable, a vector, a function, etc. For example, the coefficient of 9x² is 9. In some cases, the objects and the coefficients are indexed in the same way, leading to expressions such as

$a_1 x_1 + a_2 x_2 + a_3 x_3 + \cdots$

where a_n is the coefficient of the variable x_n for each n = 1, 2, 3, …

In a polynomial P(x) of one variable x, the coefficient of x^k can be indexed by k, giving the convention that for example

$P(x) = a_k x^k + \cdots + a_1 x^1 + a_0$

For the largest k where a_k ≠ 0, a_k is called the leading coefficient of P because most often, polynomials are written from the largest power of x, downward (i.e. x⁵ + x⁴ + x² ...).

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

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

1 is the leading coefficient of the first row, 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.

Important coefficients in mathematics include the binomial coefficients, coefficients in the statement of the binomial theorem, which can be partially found with Pascal's triangle.

In physics, many equations have coefficients associated with them. For example, $μ$ is the coefficient of friction between two objects in the equation $\textbf{F} = \mu \textbf{F}_n$ .