The concept of range has been introduced in the discussion of existence/onto. For general transformations, however, there is no similar concepts associated with uniqueness/onetoone. For a linear transformation T(x): R^{n} → R^{m}, on the other hand, the definition of uniqueness
T(x) = T(x') ⇒ x = x'.
is equivalent to
T(x  x') = 0 ⇒ x  x' = 0.
This leads to the following definition.
The discussion before the definition leads to the following conclusion.
We remark that the conclusion is comparable to the earlier observation that the following are equivalent
We also note that if T is given by a matrix A, then kernelT is all solutions of Ax = 0.
Example The system in this example has the vector (2, 2, 6) as the unique solution. Since the uniqueness is independent of the right side, we conclude that the linear transformation
T[  x_{1}  ] = [  3x_{1} + x_{2}  x_{3}  ] : R^{3} → R^{3} 
x_{2}  x_{1}  x_{2} + x_{3}  
x_{3}  2x_{1} + 2x_{2} + x_{3} 
is onetoone.
In contrast, this system has infinitely many solutions. The corresponding linear transformation
T[  x_{1}  ] = [  x_{1}  x_{2} + 3 x_{3}  ] : R^{3} → R^{3} 
x_{2}  3x_{1} + x_{2} + x_{3}  
x_{3}  x_{1} + x_{2}  x_{3} 
is not onetoone.
Finally, in this example we have a system with no solution. In general, the inconsistency of the system alone would not tell us whether the corresponding linear transformation
T[  x_{1}  ] = [  3x_{1} + x_{2}  x_{3}  ] : R^{3} → R^{3} 
x_{2}  x_{1}  x_{2} + x_{3}  
x_{3}  2x_{1}  x_{2} + x_{3} 
is onetoone or not. More computation or argument is needed. In this case, fortunately, we do not need to do any computation. By applying the basic principle of linear algebra, since 3 (number of variables/columns) = 3 (number of equations/rows), the system cannot have unique solution for any right side. Therefore T is not onetoone.
Example To determine whether the linear transformation

= 

: R^{6} → R^{4} 
is onetoone or not, we find the row echelon forms of the corresponding matrix. In this example, the matrix corresponding to T_{1}
[  1  3  2  0  1  0  ] 
1  1  1  1  0  1  
0  4  2  4  3  3  
1  3  2  2  0  0 
has been simplified, by row operations, to
[  1  3  2  0  1  0  ] 
0  2  1  1  0  1  
0  0  0  2  1  0  
0  0  0  0  0  1 
Since not all columns are pivot, we do not have uniqueness, and the transformation T is not onetoone.
In fact, we do not even need to make any computation in order to show T is not onetoone, since the solutions to the corresponding 4 equations in 6 variables cannot be unique (see the numerical consequence of uniqueness).
In order to modify the transformation to become onetoone, we need to make sure to have all rows pivot. Since the pivots columns are 1, 2, 4, 6, we delete x_{3} and x_{5} to get

= 

: R^{4} → R^{4} 
The row echelon form of the corresponding matrix is
[  1  3  0  0  ] 
0  2  1  1  
0  0  2  0  
0  0  0  1 
with all columns pivot, the transformation T_{2} is onetoone.
As in the case of onto, onetoone linear transformations also has numerical consequences. Expanding on the argument in the last example, we have