Coefficient 51203 225896603 2008-07-15T22:26:15Z 75.48.25.196 /* Linear algebra */ {{dablink|For other senses of this word, see [[coefficient (disambiguation)]].}} In [[mathematics]], a '''coefficient''' is a [[constant]] [[multiplication|multiplicative]] factor of a certain object. For example, in the expression 9''x''<sup>2</sup>, the coefficient of ''x''<sup>2 </sup> is 9. The object can be such things as a [[variable]], a [[vector (spatial)|vector]], a [[function (mathematics)|function]], etc. In some cases, the objects and the coefficients are indexed in the same way, leading to expressions such as: :<math>a_1 x_1 + a_2 x_2 + a_3 x_3 + \cdots </math> where ''a''<sub>''n''</sub> is the coefficient of the variable ''x''<sub>''n''</sub> for each ''n'' = 1, 2, 3, … In a [[polynomial]] ''P''(''x'') of one variable ''x'', the coefficient of ''x''<sup>''k''</sup> can be indexed by ''k'', giving the convention that for example: :<math>P(x) = a_k x^k + \cdots + a_1 x^1 + a_0.</math> For the largest ''k'' where ''a''<sub>''k''</sub> ≠ 0, ''a''<sub>''k''</sub> is called the '''leading coefficient''' of ''P'' because most often, polynomials are written starting from the left with the largest power of ''x''. So for example the leading coefficient of the polynomial :<math>\, 4x^5 + x^3 + 2x^2</math> is 4. The coefficients of polynomial also may be in the other order: :<math>Q(x) = a_0 x^k + a_1 x^{k-1} + \cdots + a_{k-1} x^1 + a_k</math> and must be ''a<sub>0</sub>&ne;0'' and ''a<sub>0</sub>'' is the '''leading coefficient''' of ''Q''. Important coefficients in mathematics include the [[binomial coefficient]]s which are coefficients in the statement of the [[binomial theorem]]. These can be partially found with [[Pascal's triangle]]. ==Linear algebra== In [[linear algebra]], the '''leading coefficient''' of a row in a matrix is the first nonzero entry in that row. So, for example, given :<math>M = \begin{bmatrix}1 & 2 & 0 & 6 \\ 0 & 2 & 9 & 4 \\ 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 \end{bmatrix} </math> 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.