403 – Homework 1

We are working through the argument in this note showing that every linear operator in a finite dimensional complex vector space admits an eigenvector. There are two results we need to use along the way, that you have probably seen before:

  1. Suppose that A is an n\times m-matrix (that is, n rows and m columns). The rank-nullity theorem states that  {\rm dim}({\rm null}(A))+{\rm dim}({\rm ran}(A))=m.
  2. Let V be a finite dimensional space, and let X,W be subspaces of V. The set X+W=\{x+w\mid x\in X,w\in W\} is also a subspace of V, and we have that {\rm dim}(X+W)={\rm dim}(X)+{\rm dim}(W)-{\rm dim}(X\cap W).

As your first homework set, due February 11, please write proofs of these two results.

You can work in groups, ask other people, look at notes from previous classes, consult notes or books, etc. However, please do write your own version of these arguments. Even if your own version is not as polished or complete or mathematically correct or … as what a book would have. If there are details you do not understand, please indicate so, that’s fine. I want to be able to give you meaningful feedback, and this will be pretty much meaningless and a waste of time if you simply copy someone else’s argument.

Besides writing your own version of these proofs, the only requirement I have is that what you turn in is not your preliminary or scratch work. (If you know \TeX, it would be lovely if you could type up your homework, but this is not a requirement.)

Let me know if something needs clarification.

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