# Linear algebra: Challenging problems for students by Fuzhen Zhang

By Fuzhen Zhang

Linear algebra is a prerequisite for college students majoring in arithmetic and is needed of many undergraduate and first-year graduate scholars in records, engineering, and comparable components. This absolutely up to date and revised textual content defines the discipline's major phrases, explains its key theorems, and gives over 425 instance difficulties starting from the undemanding to a few that could baffle even the main pro mathematicians. important options are highlighted before everything of every bankruptcy and a last part comprises tricks for fixing the issues in addition to ideas to every example.Based on Fuzhen Zhang's event educating and getting to know algebra over the last twenty years, Linear Algebra is the correct exam research device. scholars in starting and seminar-type complex linear algebra periods and people trying to brush up at the subject will locate Zhang's simple discussions of the subject's theories fresh and the issues assorted, fascinating, and not easy.

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W e shall therefore call a closed *-subalgebra o f a C*-algebra a C*-subalgebra. If a C*-algebra has a unit 1, then automatically | | 1 | | = 1, because ||1|| = ||1*1|| = | | 1 | | . Similarly, if p is a non-zero projection, then ||p|| = 1. If u is a unitary of A , then = 1, since ||u|| = ||u*u|| = | | 1 | | = 1. Hence, a(u) C T , for if A £

A „ ) = { ( T ( a i ) , . . , 6. a) n Show that ft(A) n M o r e precisely, set ft(A)}. Show that the canon- is a h o m e o m o r p h i s m . Let A be a unital Banach algebra. (a) If a is invertible in A , show that a(a~ ) = {A (b) For any element a £ A , show that r(a ) = (r(a)) . (c) If A is abelian, show that the Gelfand representation is isometric if and only if | | a | | = ||a|| for all a £ A . 1 n 2 - 1 | A£ cr{a)}. n 2 7. Let A be a Banach algebra. Show that the spectral radius function r: A —•> R is upper semi-continuous.

T h e o r e m . Let X be an infinite-dimensional Banach space, and suppose that w £ K(X), that u £ B(X), and that u is Fredholm. Then ind(u -f w) = ind(ii). Proof. 17, the function a: [0,1] - » Z , < h ind(u + tw), is continuous, and therefore a [ 0 , 1 ] is connected in the discrete space Z . Hence, a [ 0 , 1 ] is a singleton set, so i n d ( u ) = a(0) = a ( l ) = ind(u + w). • 1 . 4 . 3 . Remark. Let u be a Fredholm operator on an infinite-dimensional Banach space X. 18 i n d ( u ) = 0, since invertible operators are of course o f index zero.