Advanced Linear Algebra, Lecture 2.3: Algebra of linear mappings

Advanced Linear Algebra, Lecture 2.3: Algebra of linear mappings

This is somewhat of a "catch all" lecture where a number of important definitions about linear maps and basic results are given. It could have been given right after Lecture 1.1, where we originally defined linear maps, but it seems more appropriate to put in the section devoted to linear maps. The set of linear maps from X to U, denoted Hom(X,U), forms a vector space. If X=U, then they additionally define an algebra -- a vector space where we are also allowed to multiply vectors. Such linear maps are called endomorphisms, and the invertible ones form a subalgebra called the general linear group. These define similarity transformations, and an equivalence relation on Hom(X,X). We sprinkle some examples of these concepts and others throughout the lecture.

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[Galinaceo]

There seems to be a mistake at 14:20. Doesn't that proposition require X to be finite dimentional? Else the left shift and right shift operators on the space of K-sequences give a counterexample.

[gajubhat]

At 14:00, there is a simple counter example. Take the function A going from \omega to \omega n -> n+ 1 and B taking n to n -1 if n > 0 and 0 otherwise. More conditions seem to be needed.

[michaelli]

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