Note to Student: In this section we will often use , and to denote subspaces of , or any other finite-dimensional vector space, such as those we study in Vector Spaces.
Composition and Inverses of Linear Transformations
Composition of Linear Transformations
Let , and be vector spaces, and let and be linear transformations. The
composition of and is the transformation given by
Define
(You should be able to verify that both transformations are linear.) Examine the effect of on vectors of .
From the computational standpoint, the situation is simple.
This means that maps all vectors of to .
In addition to the computational approach, it is also useful to visualize what happens geometrically.
First, observe that Therefore the image of any vector of under lies on the line determined by the vector .
Even though is defined on all of , we are only interested in the action of on vectors along the line determined by . Our computations showed that all such vectors map to .
The actions of individual transformations, as well as the composite transformation are shown below.
- Proof
- Let and be linear transformations. We will show that is linear. For all vectors and of and scalars and we have:
Composition of linear transformations is associative. In other words, for linear
transformations , and
We have .
- Proof
- For all in we have:
Composition and Matrix Multiplication
In this section we will consider linear transformations of and their standard matrices.
Let and be linear transformations with standard matrices and , respectively. Then
the composite transformation has a standard matrix given by the product
.
- Proof
- For all in we have:
In Example ex:transcomp, we discussed a composite transformation given by:
Express as a matrix transformation.
We conclude this section by revisiting the associative property of matrix multiplication. At the time matrix multiplication was introduced, we skipped the cumbersome proof that for appropriately sized matrices , and , we have . (See Theorem th:propertiesofmatrixmultiplication.) 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 function composition is associative, so is matrix multiplication. We formalize this observation as a theorem.
Associativity of Matrix Multiplication Let , and be matrices of appropriate
dimensions so that the product is defined. Then
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 .
Let and be vector spaces, and let be a linear transformation. A transformation
that satisfies and is called an inverse of . If has an inverse, is called
invertible.
Let be a transformation defined by . (How would you verify that is
linear?) Show that given by 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.
- Proof
- 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.) 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 .
- Proof
- Part item:exists follows directly from the preceding discussion. Part item:unique follows from uniqueness of matrix inverses. (Theorem th:matinverseunique.)
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 Existence of the Inverse of a Linear Transformation.
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. This is because every vector in
can be expressed as a linear combination of the basis elements. The image, , can
be found by applying the linearity properties. At this point we know what
transformation does, but it is still unclear what the matrix of this linear
transformation is.
Exploration init:subtosub highlights the necessity of further discussion of inverses and of matrices
associated with linear transformations. In Example ex:subtosub, we will prove that has an
inverse, and in Example ex:inversematrixoftransform, we will address the issue of matrices associated with and
its inverse.
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.
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.
Let be a linear transformation given by
Propose a candidate for the inverse of and verify your choice using Definition
def:inverseoflintrans.