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gaussian_elimination_remarks
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| gaussian_elimination_remarks [2015/02/04 18:12] – rupert | gaussian_elimination_remarks [2016/02/11 11:18] (current) – rupert | ||
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| - | ==== Observations about Gaussian elimination ==== | ||
| - | |||
| We know that we can apply [[EROs]] to any augmented matrix into [[REF]]. | We know that we can apply [[EROs]] to any augmented matrix into [[REF]]. | ||
| Line 8: | Line 6: | ||
| * $k\le n$, because there are only $n$ rows in the whole matrix | * $k\le n$, because there are only $n$ rows in the whole matrix | ||
| - | * $k$ is precisely the number of [[leading variables]]. So $k$ is no bigger $m$, the total number of variables. | + | * $k$ is precisely the number of [[leading variables]]. So $k$ is no bigger $m$, the total number of variables; in symbols, we have $k\le m$. |
| * All the other variables are free variables, so $$ \text{$m-k$ is the number of free variables.} $$ | * All the other variables are free variables, so $$ \text{$m-k$ is the number of free variables.} $$ | ||
| Line 17: | Line 15: | ||
| For consistent systems, this shows that: | For consistent systems, this shows that: | ||
| - | * //Either// $k=m$, so $m-k=0$ | + | * //either// $k=m$; |
| - | * //or// $k<m$, so $m-k> | + | * so $m-k=0$ |
| + | * there are no free variables | ||
| + | * the system | ||
| + | * We say it has a **unique solution**. | ||
| + | * //or// $k<m$ | ||
| + | * so $m-k>0$ | ||
| + | * there is at least one free variable | ||
| + | * so the system has **infinitely many solutions** | ||
| + | * The number of free variables, $m-k$, is called the [[wp> | ||
| - | === Observation 2: systems with more variables than equations === | + | === Observation 2: systems with fewer equations |
| - | For consistent systems where $m>n$ (more variables than equations): | + | For consistent systems where $n<m$ (fewer equations |
| * $k\le n < m$, so $k< | * $k\le n < m$, so $k< | ||
| Line 28: | Line 34: | ||
| * So in this situation we //always// have infinitely many solutions. | * So in this situation we //always// have infinitely many solutions. | ||
| + | /* | ||
| === Observation 3: homogeneous systems === | === Observation 3: homogeneous systems === | ||
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| * This system is necessarily consistent, since $(0, | * This system is necessarily consistent, since $(0, | ||
| * If $n<m$, then by the previous observation there are infinitely many solutions. | * If $n<m$, then by the previous observation there are infinitely many solutions. | ||
| + | */ | ||
gaussian_elimination_remarks.1423073520.txt.gz · Last modified: by rupert