We may think of the left side of a system of linear equations as a transformation. For example, the left side of the system
x_{1}  + 2x_{2}  + 3x_{2}  =  b_{1} 
4x_{1}  + 5x_{2}  + 6x_{4}  =  b_{2} 
can be considered as the linear transformation T in the example above. By expressing vectors in vertical form, we also see that

= 

= 


In particular, the formula for the transformation T is given by a matrix A:
T(x) = Ax.
In general, for any m by n matrix A, we may construct a transformation T: R^{n} → R^{m} by the above formula. Such transformations are called matrix transformations.Example The following matrices
[  1  1  1  ] 
3  1  1  
2  2  1 
[  1  3  2  0  1  ] 
1  1  1  1  0  
0  4  2  4  3  
1  3  2  2  0 
give us matrix transformations,
T_{1}[  x_{1}  ] = [  x_{1}  x_{2} + x_{3}  ] : R^{3} → R^{3} 
x_{2}  3x_{1} + x_{2}  x_{3}  
x_{3}  2x_{1} + 2x_{2} + x_{3} 

= 

: R^{6} → R^{4} 
or written horizontally,
T_{1}: R^{3} → R^{3}, (x_{1}, x_{2}, x_{3}) → (x_{1}  x_{2} + x_{3}, 3x_{1} + x_{2}  x_{3}, 2x_{1} + 2x_{2} + x_{3})
T_{2}: R^{6} → R^{4}, (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) → (x_{1} + 3x_{2} + 2x_{3} + x_{5},  x_{1}  x_{2}  x_{3} + x_{4} + x_{6}, 4x_{2} + 2x_{3} + 4x_{4} + 3x_{5} + 3x_{6}, x_{1} + 3x_{2} + 2x_{3}  2x_{4})
S: R3^{} → R^{3}, (x_{1}, x_{2}, x_{3}) → (x_{1}  x_{2} + x_{3}, 3x_{1} + x_{2}  x_{3}  2, 2x_{1} + 2x_{2} + x_{3})
T: R^{2} → R2^{}, (x, y) → (xy, x^{2})
are not matrix transformations because they cannot be expressed as Ax for a constant matrix A.