Row space Row equivalence

the row space of matrix set of possible linear combinations of row vectors. if rows of matrix represent system of linear equations, row space consists of linear equations can deduced algebraically in system. 2 m × n matrices row equivalent if , if have same row space.

for example, matrices

(



1


0


0




0


1


1



)







and







(



1


0


0




1


1


1



)




{\displaystyle {\begin{pmatrix}1&0&0\\0&1&1\end{pmatrix}}\;\;\;\;{\text{and}}\;\;\;\;{\begin{pmatrix}1&0&0\\1&1&1\end{pmatrix}}}

are row equivalent, row space being vectors of form





(



a


b


b



)




{\displaystyle {\begin{pmatrix}a&b&b\end{pmatrix}}}

. corresponding systems of homogeneous equations convey same information:

x
=
0




y
+
z
=
0









and









x
=
0




x
+
y
+
z
=
0.






{\displaystyle {\begin{matrix}x=0\\y+z=0\end{matrix}}\;\;\;\;{\text{and}}\;\;\;\;{\begin{matrix}x=0\\x+y+z=0.\end{matrix}}}

in particular, both of these systems imply every equation of form



a
x
+
b
y
+
b
z
=
0.



{\displaystyle ax+by+bz=0.\,}