We define composition of linear transformations, inverse of a linear transformation, and discuss existence and uniqueness of inverses.

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-0030: Composition and Inverses of Linear Transformations

Composition of Linear Transformations

Let and be linear transformations. We will show that is linear. For all vectors and of and scalars and we have:

For all in we have:

Composition and Matrix Multiplication

In this section we will consider linear transformations of and their standard matrices.

For all in we have:

We conclude this section by revisiting the associative property of matrix multiplication. At the time matrix multiplication was introduced, you might have found it cumbersome to prove that for appropriately sized matrices , and , we have (See Theorem th:propertiesofmatrixmultiplication of MAT-0020). We are now in a position to prove this result with ease.

Every matrix induces a linear transformation. The product of two matrices can be interpreted as a composition of transformations. Since transformation composition is associative, so is matrix multiplication. We formalize this observation as a theorem.

Inverses of Linear Transformations

Define a linear transformation by . In other words, doubles every vector in . Now define by . What happens when we compose these two transformations? Both composite transformations return the original vector . In other words, and . We say that is an inverse of , and is an inverse of .

Linearity of Inverses of Linear Transformations

Definition def:inverseoflintrans does not specifically require an inverse of a linear transformation to be linear, but it turns out that the requirement that and is sufficient to guarantee that is linear.

The proof of this result is left to the reader. (See Practice Problem prob:inverseislinear)

Linear Transformations of and the Standard Matrix of the Inverse Transformation

Every linear transformation is a matrix transformation. (See Theorem th:matlin of LTR-0020) If has an inverse , then by Theorem th:inverseislinear, is also a matrix transformation. Let and denote the standard matrices of and , respectively. We see that and if and only if and . In other words, and are inverse transformations if and only if and are matrix inverses.

Note that if is an inverse of , then and are square matrices, and .

Part item:exists follows directly from the preceding discussion. Part item:unique follows from uniqueness of matrix inverses. (Theorem th:matinverseunique of MAT-0050)

Please note that Theorem th:existunique is only applicable in the context of linear transformations of and their standard matrices. The following example provides us with motivation to investigate inverses further, which we will do in LTR-0035.

Let Define a linear transformation by Observe that is a basis of (why?). The information about the images of the basis vectors is sufficient to define a linear transformation. (See LTR-0025) So, at this point we know what transformation does, but it is still unclear what the matrix of this linear transformation is.

Geometrically speaking, the domain of is a plane in and its codomain is .

Does have an inverse? We are not in a position to answer this question right now because Theorem th:existunique does not apply to this situation.

Exploration init:subtosub highlights the necessity of a further discussion of inverses. In Example ex:subtosub of LTR-0035, we will prove that has an inverse, and in Example ex:inversematrixoftransform of LTR-0080 we will relate the existence of an inverse of to a matrix associated with .

Practice Problems

Let and be linear transformations with standard matrices respectively. Describe the actions of , , and geometrically, as in Example ex:transcomp.
Let and be linear transformations with standard matrices respectively. Describe the actions of , , and geometrically, as in Example ex:transcomp.
Complete the Explanation of Example ex:inverseverify by verifying that .
Let be a linear transformation given by Propose a candidate for an inverse of and verify your choice using Definition def:inverseoflintrans.
Explain why linear transformation given by does not have an inverse.
Prove Theorem th:inverseislinear.
Suppose and are linear transformations with inverses and respectively. Prove that is an inverse of .