Halmos Sections 37 and 38: Matrices and Matrices of Linear Transformations(Pt. I)
Point of post: In this post I will complete the problems listed at the end of sections 37 and 38 of Halmos.
Remark: For those who are just interested in the solutions to Halmos and haven’t read my side-along postings you will probably need to see the series of posts for which this and this are the first posts for notation.
Problem: Let be defined by , and let be the canonical ordered basis . Find .
Proof: We note that for each we have that
where we’ve appealed to the convention that if . Thus
Problem: Find the matrix of the operation of the operation of conjugation on , considered as a real vector space with respect to the canonical ordered basis .
Proof: Let be the conjugation operator. We see then that
Problem: Let then compute where (where is the canonical basis for ) and .
Proof: It’s fairly easy to see that
Problem: Let and consider given by . Find where
Proof: We note that
Problem: Let be an -dimensional -space and . Let then
Under what conditions is invertible?
Proof: The claim is that is an isomorphism if and only if is an isomorphism. To see that this condition is sufficient we note that if is an isomorphism and then and thus since is an isomorphism we may conclude that . Thus, is a monomorphism, but by prior theorem we may then conclude that is an isomorphism.
To see that is a necessary condition we note that if is an isomorphism then is, in particular, an endomorphism. Thus, there exists some such that . But, the existence of a right inverse for implies that is an endomorphism, and thus since is finite dimensional we may conclude that is an isomorphism.
Problem: Prove that if and are the complex matrices
respectively. Show that , , , and .
Proof: This is purely computational, and not even cleverly so. That said, I would like to remark that this is a group under matrix multiplication which is isomorphic to the quaternions.