In this section, we discuss two of the most basic questions one can ask about a transformation: whether it is one-to-one and/or onto. For a matrix transformation, we translate these questions into the language of matrices. A transformation T
:
R
n
→
R
m
is one-to-one if, for every vector b
in R
m
,
the equation T
(
x
)=
b
has at most one solution x
in R
n
.
Another word for one-to-one is injective. Here are some equivalent ways of saying that T
is one-to-one: R
n
R
m
T
x
y
z
T
(
x
)
T
(
y
)
T
(
z
)
range
one-to-one
Here are some equivalent ways of saying that T
is not one-to-one: R
n
R
m
T
x
y
z
T
(
x
)=
T
(
y
)
T
(
z
)
range
notone-to-one
Let A be an m × n matrix, and let T ( x )= Ax be the associated matrix transformation. The following statements are equivalent:
Statements 1, 2, and 3 are translations of each other. The equivalence of 3 and 4 follows from this key observation in Section 2.4: if Ax
=
0
has only one solution, then Ax
=
b
has only one solution as well, or it is inconsistent. The equivalence of 4, 5, and 6 is a consequence of this important note in Section 2.5, and the equivalence of 6 and 7 follows from the fact that the rank of a matrix is equal to the number of columns with pivots. Recall that equivalent means that, for a given matrix, either all of the statements are true simultaneously, or they are all false. The previous three examples can be summarized as follows. Suppose that T ( x )= Ax is a matrix transformation that is not one-to-one. By the theorem, there is a nontrivial solution of Ax = 0. This means that the null space of A is not the zero space. All of the vectors in the null space are solutions to T ( x )= 0. If you compute a nonzero vector v in the null space (by row reducing and finding the parametric form of the solution set of Ax = 0, for instance), then v and 0 both have the same output: T ( v )= Av = 0 = T ( 0 ) . If T : R n → R m is a one-to-one matrix transformation, what can we say about the relative sizes of n and m ? The matrix associated to T has n columns and m rows. Each row and each column can only contain one pivot, so in order for A to have a pivot in every column, it must have at least as many rows as columns: n ≤ m . This says that, for instance, R 3 is “too big” to admit a one-to-one linear transformation into R 2 . Note that there exist tall matrices that are not one-to-one: for example, does not have a pivot in every column. A transformation T : R n → R m is onto if, for every vector b in R m , the equation T ( x )= b has at least one solution x in R n . Another word for onto is surjective. Here are some equivalent ways of saying that T is onto:
R n x T ( x ) range ( T ) R m = codomain T onto Here are some equivalent ways of saying that T is not onto:
R n x T ( x ) range ( T ) R m = codomain T notonto Let A be an m × n matrix, and let T ( x )= Ax be the associated matrix transformation. The following statements are equivalent:
Statements 1, 2, and 3 are translations of each other. The equivalence of 3, 4, 5, and 6 follows from this theorem in Section 2.3. The previous two examples illustrate the following observation. Suppose that T ( x )= Ax is a matrix transformation that is not onto. This means that range ( T )= Col ( A ) is a subspace of R m of dimension less than m : perhaps it is a line in the plane, or a line in 3 -space, or a plane in 3 -space, etc. Whatever the case, the range of T is very small compared to the codomain. To find a vector not in the range of T , choose a random nonzero vector b in R m ; you have to be extremely unlucky to choose a vector that is in the range of T . Of course, to check whether a given vector b is in the range of T , you have to solve the matrix equation Ax = b to see whether it is consistent. If T : R n → R m is an onto matrix transformation, what can we say about the relative sizes of n and m ? The matrix associated to T has n columns and m rows. Each row and each column can only contain one pivot, so in order for A to have a pivot in every row, it must have at least as many columns as rows: m ≤ n . This says that, for instance, R 2 is “too small” to admit an onto linear transformation to R 3 . Note that there exist wide matrices that are not onto: for example, does not have a pivot in every row. The above expositions of one-to-one and onto transformations were written to mirror each other. However, “one-to-one” and “onto” are complementary notions: neither one implies the other. Below we have provided a chart for comparing the two. In the chart, A is an m × n matrix, and T : R n → R m is the matrix transformation T ( x )= Ax . T isone-to-one T ( x )= b has atmostonesolution forevery b .Thecolumnsof A arelinearlyindependent. A hasapivotineverycolumn.Therangeof T hasdimension n . T isonto T ( x )= b has atleastonesolution forevery b .Thecolumnsof A span R m . A hasapivotineveryrow.Therangeof T hasdimension m . We observed in the previous example that a square matrix has a pivot in every row if and only if it has a pivot in every column. Therefore, a matrix transformation T from R n to itself is one-to-one if and only if it is onto: in this case, the two notions are equivalent. Conversely, by this note and this note, if a matrix transformation T : R m → R n is both one-to-one and onto, then m = n . Note that in general, a transformation T is both one-to-one and onto if and only if T ( x )= b has exactly one solution for all b in R m . |