Homework Notes Week 7
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1 Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we need to check that (b and (c also hold To see that (b holds suppose that A B M n n (F and a F Then aa B tr(b (aa tr(ab A a tr(b A a A B where the thrd equalty holds by lnearty of the trace so property (c holds For property (c holds we compute both A B and B A A B tr(b A (B A (B k A k k B k A k k where the fnal equalty follows because the matrx B s defned by (B k B k An dentcal calculaton gves that B A n n k A kb k and snce complex conjugaton has the followng three propertes: z + z z + z zz zz and z z for all complex numbers z z C we have that
2 B A A k B k k A k B k k A k B k k A k B k k k B k A k A B (b Use the Frobenus nner product to compute A B and A B for A ( + ( + and B By defnton X X X for all matrces X M n n (F so we star by
3 computng A A B B ( ( + A A ( ( ( + B B ( ( ( + B A ( Now we are n a poston to compute the varous nner products nvolved A A tr(a A + B B tr(b B + 4 A B tr(b A So fnally we can compute both A and B A 4 and B 4 #* Theorem Let V be an nner product space and let {v v k } be an orthogonal set n V Then for any scalars a a k : a v a v
4 Proof Wth v a descrbed as above we compute k a k v a v k j a jv j usng the lnearty n the frst varable then and conjugate-lnearty n the second (Theorem namely a v a j v j a v a j v j j j a a j v v j But v v j f j and f jtherefore for any fxed wth k we have a v a v a j v j j j a a j v v j j a a v v a v #7 Let T be a lnear operator on an nner product space V and suppose that T (x x for all x V Then T s one-to-one Proof We appeal to the fact that T s one-to-one f and only f N(T {} (Theorem 4 So suppose that x N(T e that T (x Then we must have x T (x but the only vector wth x s the zero vector so x So N(T {} and therefore T s one-to-one 4
5 # In each part apply the Gram-Schmdt process to the gven subset S of the nner product space V to obtan an orthogonal bass for span(s Then normalze the vectors to obtan an orthonormal bass for span(s and fnally compute the Fourer coeffcents of the gven vector relatve to β Fnally use Theorem 5 to verfy the result (b V R S and x To ntalze the Gram-Schmdt process we take v and then apply the next step to obtan v as v 5
6 Next we obtan v as v + The set {v v v } s an orthogonal bass so to obtan an orthonormal bass by settng w v v for then these vectors wll work specfcally { v v v v } v v s an orthonormal bass for span(s Now we compute the nner products x w for
7 x w + x w + x w Now we are supposed to verfy Theorem 5 specfcally to check that x x w w + x w w + x w w Well we can commute the rght hand sde of ths equaton drectly 7
8 x w w + x w w + x w w (d V span(s where S To start we set v v x and x and then compute v by 4 (
9 ( ( If we now normalze the vectors v v we get w Now we compute x w and x w + x w 4 4 ( ( 4 ( (7 + + x w and w 7 (( + ( ( (( + ( + 4(
10 7 ( Now agan we check that x x w w + x w w n accordance wth Theorem 5 x w w + x w w ( x # Let W span n C Fnd orthonormal bases for W and W The set s a bass for W by constructon hence by normalzng ths
11 vector we obtan an orthonormal bass of W namely x a vector y C s n W f and only f z so two such vectors are x y z and lnearly ndependent and span W so that x + z These two vectors are orthogonal Normalzng these vectors we obtan the orthonormal bass of W s a bass for W # For each oft he followng nner product space V (over F and lnear transformatons g : V F fnd a vector y such that g(x x y for all x V ( (b C C z g z z Let y z ( ( z then for any z C we have ( z z ( z + z ( z z ( z g z
12 # For each of the followng nner product space V and lnear operators T on V evaluate T at the gven vector n V (b V C T ( z z ( z Frst note that T {( ( z Let β Snce [T ] β [T ] β we have that ( z + z and x ( z ( ( z ( + for all vectors } z e the standard bass of C then [T ] β [T ] β ( ( + Now we remark that for any v C we have v [v] β so ( [ ( ] T z T z z z β [( ] [T z ] β z ( + β ( z z ( z z ( C So we can compute T on all vectors of C by the above matrx formula therefore ( ( ( + T + + ( ( + ( + ( + ( + ( + ( ( 5 +
13 #a* Theorem Let V be an nner product space and let T be a lnear operator on V Then R(T N(T Proof We wll show that followng lst of statements are all equvalent ( x N(T ( T (x ( T (x y for all y V (v x T (y for all y V (v x w for all w R(T (v x R(T so n partcular once we ve done ths we have that N(T R(T by the equvalence of ( and (v That ( and ( are equvalent s just the defnton of N(T That ( and ( are equvalent s just the observaton that the only vector u V wth u v for all v V s u That ( and (v are equvalent s just the fact that for any vectors u v V we have T (x y x T (y snce T s the adjont of T and ths property s the defnng property of the adjont The equvalence of (v and (v follows form the fact that w R(T f and only f w T (y for some y V Fnally the equvalence of (v and (v s just the defnton of R(T So ( and (v are equvalent whch means exactly that R(T N(T
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