Editing Vector space (section)

===Linear equations===
{{Main|Linear equation|Linear differential equation|Systems of linear equations}}
Systems of [[homogeneous linear equation]]s are closely tied to vector spaces.{{sfn|Lang|1987|loc = ch. VI.3.}} For example, the solutions of 
<math display=block>\begin{alignat}{9}
  && a \,&&+\, 3 b \,&\, + &\,   & c & \,= 0 \\
4 && a \,&&+\, 2 b \,&\, + &\, 2 & c & \,= 0 \\
\end{alignat}</math>
are given by triples with arbitrary <math>a,</math> <math>b = a / 2,</math> and <math>c = -5 a / 2.</math> They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. [[matrix (mathematics)|Matrices]] can be used to condense multiple linear equations as above into one vector equation, namely
<div id=equation3><math display=block>A \mathbf{x} = \mathbf{0},</math></div>
where <math>A = \begin{bmatrix}
1 & 3 & 1 \\
4 & 2 & 2\end{bmatrix}</math> is the matrix containing the coefficients of the given equations, <math>\mathbf{x}</math> is the vector <math>(a, b, c),</math> <math>A \mathbf{x}</math> denotes the [[matrix product]], and <math>\mathbf{0} = (0, 0)</math> is the zero vector. In a similar vein, the solutions of homogeneous ''linear differential equations'' form vector spaces. For example,
<div id=equation1><math display=block>f^{\prime\prime}(x) + 2 f^\prime(x) + f(x) = 0</math></div>
yields <math>f(x) = a e^{-x} + b x e^{-x},</math> where <math>a</math> and <math>b</math> are arbitrary constants, and <math>e^x</math> is the [[natural exponential function]].