### A Basic
Linear Algebra Principle

##### 2. Implication of size on existence/uniqueness

By combining the numerical implications of the existence
and the uniqueness on the size, we have

`Ax` = **b** has a unique solution for any **b**

⇒ number of rows of **A** = number of columns of **A** (**A** is a square matrix)

⇔ number of equations = number of variables

Conversely, *assume ***A** is an n by n matrix. Then we have
(if you find the general argument too difficult, try 3 by 3 matrix first)

`Ax` = **b** has solutions for any **b**

⇒ All rows are pivot
(by criterion for always existence)

⇒ Number of pivot rows is `n` (because there are `n` rows)

⇒ Number of pivot columns is `n` (by this equality)

⇒ All columns are pivot (because there are `n` columns)

⇒ The solution of a consistent system `Ax` = **b**
is unique (by criterion for uniqueness)

In other words, if the number of equations is equal to the number of variables,
then *always existence implies uniqueness*. By similar argument, we can
also prove that *uniqueness implies always existence*.

In summary, we have the following basic principle of linear algebra.

For a square matrix **A**, the following are equivalent

- Always Existence + Uniqueness
`Ax` = **b** has a unique solution for any **b**
- Always Existence
`Ax` = **b** has solutions for any **b**
- Uniqueness
- The solution of a consistent system
`Ax` = **b** is unique

Our discussion also tells us when the above happens from computational viewpoint.

For a square matrix **A**, the following are equivalent

`Ax` = **b** has a unique solution for any **b**
- All rows of
**A** are pivot
- All columns of
**A** are pivot
**A** can be row operated to become **I**

For the claim that **A** can be row operated to become **I**, please check out more details
in this exercise.

Example The system

`x`_{1} |
- `x`_{2} |
+ 2`x`_{3} |
= |
1 |

3`x`_{1} |
+ `x`_{2} |
- 2`x`_{3} |
= |
3 |

2`x`_{1} |
- `x`_{2} |
+ 2`x`_{3} |
= |
2 |

has `x`_{1} = 1, `x`_{2} = `x`_{3} = 0 as an obvious solution.
It is also easy to see that `x`_{1} = 1, `x`_{2} = 2, `x`_{3} = 1
is another solution. Therefore the system has many solutions. Since the system

`x`_{1} |
- `x`_{2} |
+ 2`x`_{3} |
= |
`b`_{1} |

3`x`_{1} |
+ `x`_{2} |
- 2`x`_{3} |
= |
`b`_{2} |

2`x`_{1} |
- `x`_{2} |
+ 2`x`_{3} |
= |
`b`_{3} |

has the same coefficient matrix, by the basic principle, it does not always have solutions.

The significance of the basic principle is the following: We may consider (always)
existence and the uniqueness as two complementary aspects of systems of linear equations.
In general, there is no relation between the two aspects. However,
*in case the size is right* (square coefficient matrix, or number of variables =
number of equations), the two aspects are equivalent.