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 is one-to-one if, for every vector in the equation has at most one solution in
Here are some equivalent ways of saying that is one-to-one:
Here are some equivalent ways of saying that is not one-to-one:
Let be an matrix, and let be the associated matrix transformation. The following statements are equivalent:
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 is a matrix transformation that is not one-to-one. By the theorem, there is a nontrivial solution of This means that the null space of is not the zero space. All of the vectors in the null space are solutions to If you compute a nonzero vector in the null space (by row reducing and finding the parametric form of the solution set of for instance), then and both have the same output:
If is a one-to-one matrix transformation, what can we say about the relative sizes of and
The matrix associated to has columns and rows. Each row and each column can only contain one pivot, so in order for to have a pivot in every column, it must have at least as many rows as columns:
This says that, for instance, is “too big” to admit a one-to-one linear transformation into
Note that there exist tall matrices that are not one-to-one: for example,
does not have a pivot in every column.
A transformation is onto if, for every vector in the equation has at least one solution in
Here are some equivalent ways of saying that is onto:
Here are some equivalent ways of saying that is not onto:
Let be an matrix, and let be the associated matrix transformation. The following statements are equivalent:
The previous two examples illustrate the following observation. Suppose that is a matrix transformation that is not onto. This means that is a subspace of of dimension less than perhaps it is a line in the plane, or a line in -space, or a plane in -space, etc. Whatever the case, the range of is very small compared to the codomain. To find a vector not in the range of choose a random nonzero vector in you have to be extremely unlucky to choose a vector that is in the range of Of course, to check whether a given vector is in the range of you have to solve the matrix equation to see whether it is consistent.
If is an onto matrix transformation, what can we say about the relative sizes of and
The matrix associated to has columns and rows. Each row and each column can only contain one pivot, so in order for to have a pivot in every row, it must have at least as many columns as rows:
This says that, for instance, is “too small” to admit an onto linear transformation to
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, is an matrix, and is the matrix transformation
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 from 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 is both one-to-one and onto, then
Note that in general, a transformation is both one-to-one and onto if and only if has exactly one solution for all in
