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--- gaussian_elimination_remarks [2015/02/04 18:12] – rupert
+++ gaussian_elimination_remarks [2016/02/11 11:18] (current) – rupert
@@ Line 1: / Line 1: @@
-==== Observations about Gaussian elimination ====
 We know that we can apply [[EROs]] to any augmented matrix into [[REF]].
@@ Line 17: / Line 15: @@
 For consistent systems, this shows that:
-  * //Either// $k=m$, so $m-k=0$ and there are no free variables, so the system, has a **unique solution**;
+  * //either// $k=m$;
-  * //or// $k<m$, so $m-k>0$ and there is at least one free free variable, so the system has **infinitely many solutions**. The number of free variables, $m-k$, is called the [[wp>dimension]] of the solution set.
+    * so $m-k=0$
+    * there are no free variables
+    * the system has one solution and no more
+    * 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** (one for each value of each free variable)
+    * The number of free variables, $m-k$, is called the [[wp>dimension]] of the solution set.
-=== Observation 2: systems with more variables than equations ===
+=== Observation 2: systems with fewer equations than variables ===
-For consistent systems where $m>n$ (more variables than equations):
+For consistent systems where $n<m$ (fewer equations than variables):
   * $k\le n < m$, so $k<m$.
@@ Line 28: / Line 34: @@
   * So in this situation we //always// have infinitely many solutions.
+/*
 === Observation 3: homogeneous systems ===
@@ Line 42: / Line 49: @@
   * This system is necessarily consistent, since $(0,0,0,\dots,0)$ is a solution.
   * If $n<m$, then by the previous observation there are infinitely many solutions.
+*/