Section4.2One-to-one and Onto Transformations¶ permalink

Objectives

Understand the definitions of one-to-one and onto transformations.

Recipes: verify whether a matrix transformation is one-to-one and/or onto.

Pictures: examples of matrix transformations that are/are not one-to-one and/or onto.

Vocabulary:one-to-one, onto.

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.

Subsection4.2.1One-to-one Transformations

Definition(One-to-one transformations)

A transformation is one-to-one if, for every vector in the equation has at most one solution in

Statements 1, 2, and 3 are translations of each other. The equivalence of 3 and 4 follows from this key observation in Section 3.1: if has only one solution, then 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 3.2, 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 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:

Wide matrices do not have one-to-one transformations

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.

Subsection4.2.2Onto Transformations

Definition(Onto transformations)

A transformation is onto if, for every vector in the equation has at least one solution in

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.

Tall matrices do not have onto transformations

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.

Subsection4.2.3Comparison

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

One-to-one is the same as onto for square matrices

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