# Abstract Nonsense

## Halmos Sections 39 and 40: Invariance and Reducibility

Point of post: In this post we complete the problems at the end of sections 39 and 40 in Halmos.

December 23, 2010

## Halmos Sections 37 and 38: Matrices and Matrices of Linear Transformations(Pt. IV)

Point of post: This is a continuation of this post.

Remark: For some strange reason the fourth (this one) and the fifth (the previous one) got mixed up in the order of posting. The number is correct, this is the fourth post in this sequence and the one preceding it the fifth.

December 19, 2010

## Halmos Sections 37 and 38: Matrices and Matrices of Linear Transformations(Pt. V)

Point of post: This is a continuation of this post.

December 19, 2010

## Halmos Sections 37 and 38: Matrices and Matrices of Linear Transformations(Pt. III)

Point of post: This is a continuation of this post.

December 19, 2010

## Halmos Sections 37 and 38: Matrices and Matrices of Linear Transformations(Pt. II)

Point of post: This is a continuation of this post.

December 19, 2010

## 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.

December 19, 2010

## Algebra Isomorphism Between Algebra of Square Matrices and the Endomorphism Algebra (Pt. I)

Point of post: In this series of posts we’ll cover a lot of ground. We’ll discuss how one can canonically associate matrices in $\text{Mat}_n\left(F\right)$ with endomorphisms on $F^n$, we’ll then discuss the ideas of ordered bases and associative unital algebra isomorphisms, and we’ll end the sequence with our main theorem which connects matrices and endomorphisms on general $n$-dimensional $F$-spaces in an interesting and instructive way.

Motivation

In our last post we saw how to endow $\text{Mat}_n\left(F\right)$ ($F^{n^2}$ in disguise) with the structure of an associative unital algebra. We hinted that besides this being another example to add to our list of associative unital algebras that it played an important, and enlightening role in the study of the endomorphism algebra of an $n$-dimensional $F$-space. To see this we’ll first show how one can canonically interpret a matrix as a linear transformation on $F^n$(in all formality what we really mean is that the square array of numbers defined as a matrix gives rise to a linear transformation which is represented and computed using the same square array of numbers). We’ll then show that this process goes in reverse. Namely, given an endomorphism on some $n$-dimensional $F$-space $\mathscr{V}$ there is a canonical way to produce a matrix. Moreover, this correspondence between linear transformations and matrices turns out to be something called an associative unital algebra isomorphism, which is just a highfalutin way of saying that the correspondence respects the operations of the domain and codomain algebras.

December 14, 2010

## Halmos Section 36: Inverses (Pt. III)

Point of post: This is a continuation of this post.

December 4, 2010

## Halmos Section 36: Inverses (Pt. II)

Point of post: This is a continuation of this post.

December 3, 2010

## Halmos Section 36: Inverses (Pt. I)

Point of post: In this post we complete the problems at the end of section 36 in Halmos.

December 2, 2010