It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out.
I am opposed to textbooks... I find it hateful to give a course where I have to plough my way through chapter after chapter of a given book. The liveliness of the lecture, which is meant to give an impetus to the sudents, would suffer tremendously.
This skipping is another important point. It should be done whenever a proof seems too hard or whenever a theorem or a whole paragraph does not appeal to the reader. In most cases he will be able to go on and later he may return to the parts which he skipped.
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