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Section4.2One-to-one and Onto Transformations

Objectives
  1. Understand the definitions of one-to-one and onto transformations.
  2. Recipes: verify whether a matrix transformation is one-to-one and/or onto.
  3. Pictures: examples of matrix transformations that are/are not one-to-one and/or onto.
  4. Vocabulary words: 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 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 .

Here are some equivalent ways of saying T is one-to-one:”

  • The transformation T takes different vectors in R n to different vectors in R m .
  • Different inputs of T have different outputs.
  • If T ( u )= T ( v ) then u = v .

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 T is not one-to-one:”

  • There exists some vector b in R m such that the equation T ( x )= b has more than one solution x in R n .
  • There are two different inputs of T with the same output.
  • There exist vectors u , v such that u A = v but T ( u )= T ( v ) .

R n R m T x y z T ( x )= T ( y ) T ( z ) range notone-to-one
Proof

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 ) .

Wide matrices do not have one-to-one transformations

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,

AEC 100010000000 BFD

does not have a pivot in every column.

Subsection4.2.2Onto Transformations

Definition(Onto transformations)

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 .

Here are some equivalent ways of saying T is onto:”

  • The range of T is equal to the codomain of T .
  • Every vector in the codomain is the output of some input vector.

R n x T ( x ) range ( T ) R m = codomain T onto

Here are some equivalent ways of saying T is not onto:”

  • The range of T is smaller than the codomain of T .
  • There exists a vector b in R m such that the equation T ( x )= b does not have a solution.
  • There is a vector in the codomain that is not the output of any input vector.

R n x T ( x ) range ( T ) R m = codomain T notonto
Proof

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.

Tall matrices do not have onto transformations

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,

G 1 12 22 4 H

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, 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. T isonto T ( x )= b has atleastonesolution forevery b .Thecolumnsof A span R m . A hasapivotineveryrow.
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 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 .