Exercise 2.6.14. Suppose that , , and are linear transformations, with taking vectors from to , taking vectors from to , and taking vectors from to . Consider the product of these transformations. It starts with a vector in and produces a new vector in . It then follows 2V on page 123 to apply the product linear transformation to producing a final vector in representing the result .

i) Is the result of this process the same as separately applying followed by followed by ?

ii) Is the final result of this process the same as applying followed by ? In other words, does the associative law apply to linear transformations, so that ?

Answer: i) In computing we start with a vector in and apply the linear transformation to produce a new vector in . We then apply the product transformation to the vector to produce a vector in .

But by rule 2V the linear transformation is simply the composition of the two linear transformations and . So applying the product transformation to the vector to produce a vector in is equivalent to first applying the transformation to the vector to produce a vector in and then applying the transformation to the vector to produce a vector in . In other words, .

So the final result is the same whether we apply to or apply to and then apply to . Since the first step, namely applying to to produce , is the same in both cases, the result of (applying and then ) is the same as applying first , then , and then . In other words, .

ii) Just as and can be composed together to form a linear transformation from into , so can and can be composed together to create a linear transformation from into . By rule 2V we know that this is the same as applying to a vector in to produce a vector in and then applying to to produce a vector in . In other words, .

In either case we can then take the resulting vector and transform it using to produce a vector in . In other words, . But from the previous answer we also know that . From the two equations together we see that and thus that the linear transformations , and obey the law of associativity.

BONUS: As noted in the text, associativity of linear transformations implies associativity of matrix multiplication.

Suppose that we choose sets of basis vectors in the vector spaces , , , and . (Our choice is arbitrary; any basis sets will do.) We can then represent the transformation by a matrix constructed using the chosen basis vectors in and . We can represent the transformation by a matrix constructed using the chosen basis vectors in and . Finally, we can represent the transformation by a matrix constructed using the chosen basis vectors in and .

By rule 2V composition of two linear transformations creates a new linear transformation that can be represented by a matrix that is the product of the matrices representing the original linear transformations. Since we have we then also have . Any matrix can represent a linear transformation, so we then have for any three matrices , , and that can be multiplied together.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.