a matrix is said to be in row echelon form if:
a.) all of the leading coefficients is 1.
b.) each of the succeeding row from the bottom has the leading coefficient of 1
c.) if the bottom row has all zeros (consistent dependent system).
 |
| an example of a matrix in row-echelon form. |
a matrix is said to be in reduced row-echelon form if:
a.) it is in row-echelon form.
b.) every leading coefficient is 1 and is the only non zero entry. (besides the constant).
 |
| an example of a matrix in reduced row-echelon form. |
![\left[\begin{array}{rrr|r}
1 & 3 & 1 & 9 \\
1 & 1 & -1 & 1 \\
3 & 11 & 5 & 35
\end{array}\right]\to
\left[\begin{array}{rrr|r}
1 & 3 & 1 & 9 \\
0 & -2 & -2 & -8 \\
0 & 2 & 2 & 8
\end{array}\right]\to
\left[\begin{array}{rrr|r}
1 & 3 & 1 & 9 \\
0 & -2 & -2 & -8 \\
0 & 0 & 0 & 0
\end{array}\right]\to
\left[\begin{array}{rrr|r}
1 & 0 & -2 & -3 \\
0 & 1 & 1 & 4 \\
0 & 0 & 0 & 0
\end{array}\right]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uG0-0Z4yP1m8rhJDXm8ZIoY-aVsnm3C1uuiCaHC9PHXObGBk8G59zE0lg2eIkMJ0gmHGXAWvD2iiwHttgCIYw73JpYwR2cNbGKc0SG7sIfX9Ay5V9VWK_MqPqDF3KKmgiC-nQMtcOF7jy2v9gVjg=s0-d)
over a field 
, and so on).
. While this can, in theory, be solved using a matrix inverse
over a field 
has the structure of a ring with all the usual axioms for an inner addition and an inner multiplication together with distributive laws, therefore giving it more structure than a ring. A linear algebra also admits an outer operation of multiplication by scalars (that are elements of the underlying field 
). For example, the set of all linear transformations from a vector space 
to itself over a field 
forms a linear algebra over 
. Another example of a linear algebra is the set of all real square matrices over the field 
of the real numbers.