Differences

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--- row_echelon_form [2015/01/29 11:27] – created rupert
+++ row_echelon_form [2016/01/28 11:27] (current) – rupert
@@ Line 1: / Line 1: @@
 A matrix is in **row echelon form**, or **REF**, if it has all of the following three properties:
-  - The zero rows of the matrix (if any) are all at the bottom of the matrix.
+  - The [[zero rows]] of the matrix (if any) are all at the bottom of the matrix.
-  - In every non-zero row of the matrix, the leading entry is $1$.
+  - In every non-zero row of the matrix, the [[leading entry]] is $1$.
   - If row $i$ and row $(i+1)$ are both non-zero, then the leading entry in row $(i+1)$ is to the right of the leading entry in row $i$. <html><br /></html>In other words, as you go down the rows, the leading entries must go to the right.
+For example, $\left[\begin{smallmatrix} 1&2&3&4&5\\0&1&2&3&4\\0&0&1&2&3\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 1&2&3&4&5\\0&1&2&3&4\\0&0&1&2&3\\0&0&0&0&0\end{smallmatrix}\right]$ are both in REF, but
+  - $\left[\begin{smallmatrix} 1&2&3&4&5\\0&0&0&0&0\\0&0&1&2&3\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 1&2&3&4&5\\0&0&0&0&0\\0&0&1&2&3\\0&0&0&0&0\end{smallmatrix}\right]$ are not in REF, since they each have a zero row which isn't at the bottom;
+  - $\left[\begin{smallmatrix} 1&2&3&4&5\\0&2&3&4&1\\0&0&1&2&3\end{smallmatrix}\right]$ is not in REF, since the leading entry on the second row isn't $1$;
+  - $\left[\begin{smallmatrix} 0&1&2&3&4\\1&2&3&4&5\\0&0&1&2&3\end{smallmatrix}\right]$ is not in REF, since the leading entry in row $2$ is not to the right of the leading entry in row $1$.