The text Basic Linear Algebra by Blyth and Robertson is a splendid introduction to the subject. It is published in the Springer SUMS series and the ISBN is 3-540-76122-5. This text gives a direct and short proof of the Cayley-Hamilton theorem, via some cunning matrix algebra. However, I have written some notes which get to the decomposition into generalized eigenspaces very fast using a road less travelled. You can read off the Cayley-Hamilton theorem and the characterization of diagonalizability in terms of the minimum polynomial at leisure. The formats are dvi, postscript and pdf.

Basic Linear Algebra (Blyth and Robertson)

• p. 85 The proof of Corollary 1 is broken. It assumes that I is finite. At this stage of the theory we must allow for the (aburd) possibility that a finite dimensional vector space can contain an infinite linearly independent subset because we have yet to exclude this possibility. A `fix' is to insist that I is finite in Corollary 1.

Alternatively, look back to Theorem 5.7 on page 81. We almost eliminate the problem there. This Theorem shows that no finite linearly independent subset of an n-dimensional space can have more than n elements. However, there remains the possibility that there is an infinite linearly independent subset. However, an infinite linearly independent subset would have a subset of size n+1 and so cannot exist. This is another Corollary to Theorem 5.7 and if you take it on board, the problem with Corollary 1 on p. 85 goes away.

• p. 90 Question 5.37. The problem with the original wording is that set notation supresses repetitions, so for example {f_1, f_1} is linearly independent since {f_1, f_1} = {f_1}. To fix the problem we can talk about a finite sequence of functions f_1 to f_n instead of a set of functions.
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