Let us consider three examples.

First Example The general solution of a system of linear equations

x_{1} |
+ x_{2} |
- 5x_{4} |
= | 3 | |

x_{3} |
+ 2x_{4} |
= | 4 |

is usually written as expressions of nonfree variables (say `x`_{1}, `x`_{3}) in terms of free variables (say `x`_{2}, `x`_{4}). The solution can also be written in the vector form

= [x |
x_{1} |
] = [ | 3 - x_{2} + 5x_{4} |
] = [ | 3 | ] + x_{2}[ |
-1 | ] + x_{4}[ |
5 | ] =
+ uc_{1}v_{1}
+ c_{2}v_{2}, |

x_{2} |
x_{2} |
0 | 1 | 0 | ||||||

x_{3} |
4 - 2x_{4} |
4 | 0 | -2 | ||||||

x_{4} |
x_{4} |
0 | 0 | 1 |

where ** u**,

Second Example Consider a differential equation

`f`'' - 4`f`' + 3`f` = 3`t` - 1, for differentiable functions `f` = `f`(`t`).

The general solution is

`f` = (`t` + 1) + `c`_{1}`e ^{t}` +

where `u`(`t`), `v`_{1}(`t`), `v`_{2}(`t`) are fixed functions, and `c`_{1}, `c`_{2} are arbitrary constants.

Third Example Consider a matrix equation (see this exercise)

[ | 1 | -2 | 3 | ] = X, for 2 by 3 matrices I.X |

-2 | 3 | -4 |

The general solution is

= [X |
1 | -2 | ] + c_{1}[ |
1 | 0 | ] + c_{2}[ |
0 | 1 | ] =
+ Uc_{1}V_{1}
+ c_{2}V_{2}, |

0 | 0 | 2 | 0 | 0 | 2 | ||||

0 | 0 | 1 | 0 | 0 | 1 |

where ** U**,

Although the three problems are about different kinds of objects (euclidean vectors, differentiable functions, matrices), the solutions have similar structures. This suggests something common among these objects, which we will see is the linearity of the left side of the equations.

The purpose of this chapter is to explore the similarity illustrated by the three examples. The similarity enables us to extend the earlier discussion about euclidean vectors (such as systems of linear equations, linear transformations, existence, uniqueness, etc.) to much wider context.