I'm using this book as an undergraduate, so my rating is clearly skewed, as evidenced by the huge "Graduate Texts in Mathematics" on the cover. We've only covered the first five chapters so far, and while the overarching ideas are quite clear, I find the notation confusing. No (even small) reviews of the linear algebra you studied years ago; it just dives in. Perhaps graduate students can follow it all quickly with no concrete examples, but it takes me a few readings through each section to begin to understand what is being said. By concrete I mean a real-world see-it put-your-hands-on-it example, or at least an example involving a few numbers as elements.

Here is an excerpt (so you can judge for yourself how helpful the first chapter will be for you) of a representation example from section 1.2 entitled 'Basic Examples:' "Leg g be the order of G, and let V be a vector space of dimension g, with a basis (e-sub-t)sub-t-in-g indexed by the elements t of G. For s-in-G, let rho-sub-s be the linear map of V into V which sends e-sub-t to e-sub-st; this defines a linear representation, which is called the regular representation of G. Its degree is equal to the order of G. Note that e-sub-s = rho-sub-s(e-sub1); hence note that the images of e-sub1 form a basis of V. Conversely, let W be a representation of G containing a vector w such that the rho-sub-s(w), s-in-G, form a basis of W; then W is isomorphic to the regular representation (an isomorphism tau: V --> W is defined by putting tau(e-sub-s) = rho-sub-s(w))."

The language is very concise and usually quite clear, and I suppose for someone with a sophisticated math background it could be a preferred book. For someone like me who has had only one semester of introductory linear algebra two years ago, I would prefer a more "bridging" text -- that is, one which often and quickly reviewed basic concepts from linear algebra and was less concise in its explanations of definitions and examples.

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