We prove that a linear transformation has an inverse if and only if the transformation is “one-to-one” and “onto”.

Note to Student: In this module we will often use , and to denote the domain and codomain of linear transformations. If you are familiar with abstract vector spaces, you can regard , and as finite-dimensional vector spaces, otherwise you may think of , and as subspaces of .

LTR-0035: Existence of the Inverse of a Linear Transformation

In Exploration ep:inverse of LTR-0030 we examined a linear transformation that doubles all input vectors, and its inverse , that halves all input vectors. We observed that the composite functions and are both identity transformations. Diagrammatically, we can represent and as follows:

This gives us a way of thinking about an inverse of as a transformation that “undoes” the action of by “reversing” the mapping arrows. We will now use these intuitive ideas to understand which linear transformations are invertible and which are not.

Given an arbitrary linear transformation , “reversing the arrows” may not always result in a transformation. Recall that transformations are functions. The figures below show two ways in which our attempt to “reverse” may fail to produce a function.

First, if two distinct vectors and map to the same vector in , then reversing the arrows gives us a mapping that is clearly not a function.

Second, observe that our definition of an inverse of requires that the domain of the inverse transformation be . (Definition def:inverse, LTR-0030) If there is a vector in that is not an image of any vector in , then cannot be in the domain of an inverse transformation.

We now illustrate these potential issues with specific examples.

Our next goal is to develop vocabulary that would allow us to discuss issues illustrated in Figures and .

One-to-one Linear Transformations

Figure gave us a diagrammatic representation of a transformation that maps two distinct elements, and to the same element , making it impossible for us to “reverse the arrows” in an attempt to find the inverse transformation. Based on this example, it is reasonable to conjecture that for a transformation to be invertible, the transformation must be such that each output is the image of exactly one input. Such transformations are called one-to-one.

The transformation in Figure is not one-to-one because and map to the same vector , (i.e. ), yet the diagram suggests that .

Since transformation in Example ex:notonto is one-to-one but not invertible we can conjecture that being one-to-one is a necessary, but not a sufficient condition for a linear transformation to have an inverse. We will consider the other necessary condition next.

“Onto” Linear Transformations

Figure makes a convincing case that for a transformation to be invertible every element of the codomain must have something mapping to it. Transformations such that every element of the codomain is an image of some element of the domain are called onto.

Existence of Inverses

We will first assume that is one-to-one and onto, and show that there exists a transformation such that and . Because is onto, for every in , there exists in such that . Moreover, because is one-to-one, vector is the only vector that maps to . To stress this, we will say that for every , there exists such that . (Since every maps to exactly one , this notation makes sense for elements of as well.) We can now define by . Then We conclude that and . Therefore is an inverse of .

We will now assume that has an inverse and show that must be one-to-one and onto. Suppose then but then We conclude that is one-to-one.

Now suppose that is in . We need to show that some element of maps to . Let . Then We conclude that is onto.

Uniqueness of Inverses

Definition def:inverse of LTR-0030 refers to as an inverse of , implying that there may be more than one such transformation . We will now show that if such a transformation exists, it is unique. This will allow us to refer to the inverse of and to start using to denote the unique inverse of .

Let be a linear transformation. If is an inverse of , then satisfies

Suppose there is another transformation, , such that We now show that .

Practice Problems

Show that a linear transformation with standard matrix is not one-to-one.
Show that multiple vectors map to .
Show that a linear transformation with standard matrix is not onto.
Find such that has no solutions.
Suppose that a linear transformation has a standard matrix such that .

Prove that is one-to-one.

How many solutions does have?
Prove that is onto.
Does have a solution for every ?
Define a transformation by Show that is a linear transformation that has an inverse.
You will need to demonstrate that is one-to-one and onto.
Let . Define a linear transformation by Prove that has an inverse.