solutions3.3 - Math 307 Problems for section 3.3 3i 1 4 1...

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Math 307: Problems for section 3.31.Verify that the vectors q1=154and q2=Compute the coefficientsc1andc2in the expansion3i1543iform an orthonormal basis forC2.
2.Show that for v,wCnkv+wk2=kvk2+kwk2+ 2 Re(hv,wi)and use this to prove the polarization identityhv,wi=14kv+wk2- kv-wk2+ikv-iwk2-ikv+iwk2This identity shows that we can compute the inner product using the norm.Use itto show that ifQis a matrix preserving lengths, i.e.,kQxk=kxkfor every xCn,thenhQv, Qwi=hv,wifor every v,wCn.SincehQv, Qwi=hv, Q*Qwithis implies thathv, Q*Qwi=hv,wifor every v,wCn.This in turn shows thatQ*Q=I, i.e., thatQisunitary.kv+wk2=hv+w,v+wi=hv,vi+hv,wi+hw,vi+hw,wi=kvk2+hv,wi+hv,wi+kwk2=kvk2+kwk2+ 2 Re(hv,wi)
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Thuskv+wk2- kv-wk2=kvk2+kwk2+ 2 Re(hv,wi)-(kvk2+kwk2-2 Re(hv,wi))= 4 Re(hv,w,)iand using this equality withwreplaced by-iwyieldskv-iwk2- kv+iwk2= 4 Re(hv,-iwi)= 4 Re(-ihv,wi)= 4 Im(hv,wi)The last two equations give the polarization identity.IfkQxk=kxkfor everyxCn, then using the polarization identity we find thathQv, Qwi=14kQv+Qwk2- kQv-Qwk2+ikQv-iQwk2-ikQv+iQwk2=14kQ(v+w)k2- kQ(v-w)k2+ikQ(v-iw)k2-ikQ(v+iw)k2=14kv+wk2- kv-wk2+ikv-iwk2-ikv+iwk2=hv,wi3.Show that if q1,q2, . . . ,qnform a basis inRn, then they also form a basis when regarded asvectors inCn. In other words, show that 1.) if the only linear combinationc1q1+· · ·+cnqnusing real numbersc1, . . . , cnthat equals zero hasc1=· · ·=cn= 0, then the same is true forcomplex numbers, and 2.) if every vector inRncan be written asc1q1+· · ·+cnqnfor somereal numbersc1, . . . , cnthen every vector inCncan be written as a linear combination usingcomplex numbers. If the basis q1,q2, . . . ,qnis orthonormal inRnis is also orthonormal inCn?Suppose a complex linear combination ofq1,q2, . . . ,qnequals zero. Writing the complex scalars asck+idkwe have(c1+id1)q1+· · ·+ (cn+idn)qn=0

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