Describe the mistake
For exercise 2.18, the linear maps f and g should just be linear transformations, not automorphisms. If f and g are automorphisms they're bijective, so their kernels are {0} and the question becomes trivial. For example in ker(f) = ker(g dot f) everything just cancels to the set {0} and there's no need to use the condition that g dot f = id_E.
Location
version
Draft (2023-02-15) of “Mathematics for Machine Learning”.
Chapter
chapter 2 exercises
page
Page 68 in the ebook
Proposed solution
Change "Let f and g be two automorphisms on E" to "Let f and g be two endomorphisms on E."
Describe the mistake For exercise 2.18, the linear maps f and g should just be linear transformations, not automorphisms. If f and g are automorphisms they're bijective, so their kernels are {0} and the question becomes trivial. For example in ker(f) = ker(g dot f) everything just cancels to the set {0} and there's no need to use the condition that g dot f = id_E.
Location
Proposed solution Change "Let f and g be two automorphisms on E" to "Let f and g be two endomorphisms on E."