In this section, we study compositions of transformations. As we will see, composition is a way of chaining transformations together. The composition of matrix transformations corresponds to a notion of multiplying two matrices together. We also discuss addition and scalar multiplication of transformations and of matrices.
Composition means the same thing in linear algebra as it does in Calculus. Here is the definition.
Let and be transformations. Their composition is the transformation defined by
Composing two transformations means chaining them together: is the transformation that first applies then applies (note the order of operations). More precisely, to evaluate on an input vector first you evaluate then you take this output vector of and use it as an input vector of that is, Of course, this only makes sense when the outputs of are valid inputs of that is, when the range of is contained in the domain of
Here is a picture of the composition as a “machine” that first runs then takes its output and feeds it into there is a similar picture in this subsection in Section 3.1.
Recall from this definition in Section 3.1 that the identity transformation is the transformation defined by for every vector
Let be transformations and let be a scalar. Suppose that and that in each of the following identities, the domains and the codomains are compatible when necessary for the composition to be defined. The following properties are easily verified:
The final property is called associativity. Unwrapping both sides, it says:
In other words, both and are the transformation defined by first applying then then
Composition of transformations is not commutative in general. That is, in general, even when both compositions are defined.
In this subsection, we introduce a seemingly unrelated operation on matrices, namely, matrix multiplication. As we will see in the next subsection, matrix multiplication exactly corresponds to the composition of the corresponding linear transformations. First we need some terminology.
Let be an matrix. We will generally write for the entry in the th row and the th column. It is called the entry of the matrix.
Let be an matrix and let be an matrix. Denote the columns of by
The product is the matrix with columns
In other words, matrix multiplication is defined column-by-column, or “distributes over the columns of ”
In order for the vectors to be defined, the numbers of rows of has to equal the number of columns of
If has only one column, then also has one column. A matrix with one column is the same as a vector, so the definition of the matrix product generalizes the definition of the matrix-vector product from this definition in Section 2.3.
If is a square matrix, then we can multiply it by itself; we define its powers to be
Recall from this definition in Section 2.3 that the product of a row vector and a column vector is the scalar
The following procedure for finding the matrix product is much better adapted to computations by hand; the previous definition is more suitable for proving theorems, such as this theorem below.
Let be an matrix, let be an matrix, and let Then the entry of is the th row of times the th column of
Here is a diagram:
Although matrix multiplication satisfies many of the properties one would expect (see the end of the section), one must be careful when doing matrix arithmetic, as there are several properties that are not satisfied in general.
While matrix multiplication is not commutative in general there are examples of matrices and with For example, this always works when is the zero matrix, or when The reader is encouraged to find other examples.
The point of this subsection is to show that matrix multiplication corresponds to composition of transformations, that is, the standard matrix for is the product of the standard matrices for and for It should be hard to believe that our complicated formula for matrix multiplication actually means something intuitive such as “chaining two transformations together”!
Let and be linear transformations, and let and be their standard matrices, respectively, so is an matrix and is an matrix. Then is a linear transformation, and its standard matrix is the product
The theorem justifies our choice of definition of the matrix product. This is the one and only reason that matrix products are defined in this way. To rephrase:
The matrix of the composition of two linear transformations is the product of the matrices of the transformations.
Recall from this definition in Section 3.3 that the identity matrix is the matrix whose columns are the standard coordinate vectors in The identity matrix is the standard matrix of the identity transformation: that is, for all vectors in For any linear transformation we have
and by the same token we have for any matrix we have
Similarly, we have and
In this subsection we describe two more operations that one can perform on transformations: addition and scalar multiplication. We then translate these operations into the language of matrices. This is analogous to what we did for the composition of linear transformations, but much less subtle.
To emphasize, the sum of two transformations is another transformation called its value on an input vector is the sum of the outputs of and Similarly, the product of with a scalar is another transformation called its value on an input vector is the vector
Let be transformations and let be scalars. The following properties are easily verified:
In one of the above properties, we used to denote the transformation that is zero on every input vector: for all This is called the zero transformation.
We now give the analogous operations for matrices.
Let be linear transformations with standard matrices respectively, and let be a scalar.
In view of the above fact, the following properties are consequences of the corresponding properties of transformations. They are easily verified directly from the definitions as well.
Let be matrices and let be scalars. Then:
In one of the above properties, we used to denote the matrix whose entries are all zero. This is the standard matrix of the zero transformation, and is called the zero matrix.
We can also combine addition and scalar multiplication of matrices with multiplication of matrices. Since matrix multiplication corresponds to composition of transformations (theorem), the following properties are consequences of the corresponding properties of transformations.
Let be matrices and let be a scalar. Suppose that is an matrix, and that in each of the following identities, the sizes of and are compatible when necessary for the product to be defined. Then:
Most of the above properties are easily verified directly from the definitions. The associativity property however, is not (try it!). It is much easier to prove by relating matrix multiplication to composition of transformations, and using the obvious fact that composition of transformations is associative.
