1. Let C denote the field of complex numbers. As with any field, we can consider vector spaces, linear transformations, and matrices over C rather than over our usual field R. (a) The complex numbers C can be viewed as a vector space over either C or R with the usual scalar multiplication. Prove that C has dimension 1 as a vector space over C but has dimension 2 as a vector space over R. In each case, give an explicit basis.

(b) Diagonalize the following matrices over C by giving QA, QB ? M22(C) so that Q ?1 A AQA and Q ?1 B BQ are diagonal. A = 0 1 ?1 0 , B = 1 1 ?1 0

7. Suppose that A ? Mnn(R) has two distinct eigenvalues ?1 and ?2 with dim(E?1 ) = n ? 1. Prove that A is diagonalizable.

And Part b and d in jpg. plz show detailed and clearly works. Make sure correct!! thanks

3. For their; (ination, are magir choose arbitraryr matrix mpruaantatiaa. usually use the standard basis, and do the same as what we did in the previcrus EIGI'CiSE. Ha hm we'll have [T13 = D and the set of minim: VH'JSDI'S of Q is the Drdmed basis ,8 {3] It's not diagn-nalizahlc since dim[Eg} is 1 but not 4. Hill -ll{.'| [-3) It's not diagonalimhla since its dimisc polynomial rims not split. ..1 [I [I l l {I {h]1t'sdiaganassab1awith= u 1 0 me: u u 1. 1 l l l {:1} It's diagnnalizahla 1with 13- (: 2 [I] and Q =(= l l l)_ [I [a ]| It's diagonamable with D: ( 1 i] [f] 111'; tagnnaligahle with D - [ 1': i] J 'I'J.T_ __J. L.-._-- -__:_ 11__1-__ 4.1.- -L-___J.__JT' L- d.L_ -1.-___|._ I. :. J.L_ I. ProblI-s L Let C denote the eld of complex numbers. As with any eld, we can consider vector spaces, linear transformations, and matrices over '33 rather than over our usual eld H. (a) The complex numbers '13 can be viewed as a vector space over either {3 or R with the usual scalar multiplication. Prove that [C has dimension 1 as a vector space cwer C but has dimension 2 as a vector space over IL In each caseI give an explicit basis. [b] Diagonalize the following matrices over {I by giving (.14, (.23 E MMEC} so that leac and (25' HQ are diagonal. em). 34.1.3.) . Section 5.1 Problem 1. No justication is required. Section 5.2 Problem 1. No justication is required. Section 5.2 Problem 2[d},[f}. Section 5.2 Problem 3(b}.[d}. Prove that similar matrices have the same characteristic polynomial. \"FY-\"PW\" . Suppose that A E MnxER} has two distinct eigenvalues A1 and A2 with dim{EM} = n 1. Prove that A is diagonalieable. 3. Sect-ion 5.3 Problem 1. No justication is required. 9. Section 5.3 Problem 6