RSI at MIT Summer 2002The text Matrix Theory and Linear Algebra by I. N. Herstein and David J. Win-ter (Macmillan, 1988, ISBN 0-02-353951-8) is excellent. For those new to the sub-ject, or for those interested in applications to physics, there are some minor pointsthat might need further clarification.Page 55, Defintion of Inner Product: This definition is fine, and is usedconsistently in the text. However, other texts (primarily physics texts) use a differ-ent but entirely equivalent definition, specifically(v, w)=¯αγ +¯βδ.It’s important to realize that this is a matter of convention, and will not affect anystated results. Those accustomed to reading this or similar texts (or physics texts)will make the necessary adjustment automatically. A useful but tedious exercise isto repeat all derivations using the above convention. One item that is different inthe above convention is part (4) of Lemma 1.12.1, which would be(σv, w)=¯σ (v, w)=(v, ¯σw) .It’s straightforward to show that with the above convention(v, σw)=σ (v, w)=(¯σv, w) .The fact is, any result involving the inner product will not depend on the conventionused. For example, try proving Theorem 1.12.3 using the alternate convention forthe inner product. (If this is physics, we’re playing with a stacked deck.)Page 9, Problem 16: This is a good place to look ahead. Try this problem(not a hard one) using the notation introduced in Section 1.3.Page 9, Problems 20 & 21: You’ll eventually see an easier way of doingProblem 21 (a generalization of Problem 20). If you have the time, try doingProblem 21 using the same techniques used to solve Problem 20.Page 13, Problem 5: This indeed is a bit of number-crunching, but it’sworth doing, if only to appreciate the easier way that will be introduced later.Page 13, Problem 5: Strictly speaking, this problem must be solved byinduction on n. These sorts of 2 × 2 matrices appear so frequently that you mayend up recognizing the results immediately.1Page 15, Problem 21: Similarly, this should be done by induction. Thematrix A is a specific form of a class of matrices that you will see frequently thissummer.Pages 19-21: Note the instruction at the beginning of this set of problems, tothe effect that summation notation should be use. You are encouraged to try anyproblems for the specific case where any matrices are elements of M2(R).Page 20, Problem 11: For instance, this problem turns out to be far easier(that’s a judgement call, of course) using the summation notation; try it both ways.ErrataLike all texts that contain lots of math, some transcription errors can arise. Aswith all good texts, these errors are few. Here are some.14, Problem 18: Should beShow that if C = ABA−1=abcd, ... .Page 17, Proof of Corillary 1.4.3: Missing parenthesis: Should betr A−1(BA)=tr (BA) A−1=tr(B) .Page 46, First line of text: Should be “How do the elements of C ...”(that should be the “C” that represents the set of matrices defined on Page 45, afont that is hard to reproduce).Page 56, Proof of Theorem 1.12.4, second line of text (following firstdisplaye d equation): As in the stated theorem, the result is(v, w)=(w, v).If you have lots of free time, show that this is true if the alternate convention,(v, w)=¯αγ +¯βδ,is used
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