Editing Coefficient (section)

==Linear algebra==
In [[linear algebra]], a [[system of linear equations]] is frequently represented by its [[coefficient matrix]].  For example, the system of equations
<math display="block">
\begin{cases} 
2x + 3y = 0 \\
5x - 4y = 0
\end{cases},</math>
the associated coefficient matrix is <math>\begin{pmatrix}
2 & 3 \\
5 & -4
\end{pmatrix}.
</math>  Coefficient matrices are used in algorithms such as [[Gaussian elimination]] and [[Cramer's rule]] to find solutions to the system.

The '''leading entry''' (sometimes ''leading coefficient''{{cn|date=March 2022}}) of a row in a matrix is the first nonzero entry in that row. So, for example, in the matrix
<math display="block">
\begin{pmatrix}
1 & 2 & 0 & 6\\
0 & 2 & 9 & 4\\
0 & 0 & 0 & 4\\
0 & 0 & 0 & 0
\end{pmatrix},
</math>
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 [[constant (mathematics)|constants]] in elementary algebra, they can also be viewed as variables as the context broadens. For example, the [[coordinates]] <math>(x_1, x_2, \dotsc, x_n)</math> of a [[vector (geometric)|vector]] <math>v</math> in a [[vector space]] with [[basis (linear algebra)|basis]] <math>\lbrace e_1, e_2, \dotsc, e_n \rbrace </math> are the coefficients of the basis vectors in the expression 
<math display="block"> v = x_1 e_1 + x_2 e_2 + \dotsb + x_n e_n .</math>