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Forum Studenti •Simulazione scritto d'esame 4 - Pagina 2
Pagina 2 di 3

Re: Simulazione scritto d'esame 4

Inviato: martedì 3 giugno 2014, 9:18
da Massimo Gobbino

Re: Simulazione scritto d'esame 4

Inviato: martedì 3 giugno 2014, 12:16
da GIMUSI
con Sylvester si impara sempre qualcosa...non avevo mai pensato di impiegarlo anche per determinare i sottospazi nei quali la forma è definita positiva...proverò a svolgere qualche esempio per vedere se ho afferrato l'idea :)

Re: Simulazione scritto d'esame 4

Inviato: martedì 3 giugno 2014, 13:23
da Gabe
Avevo intuito che non andasse bene sempre...però non sapevo in quali casi!
Per quanto riguarda il completamento dei quadrati avete qualche consiglio/suggerimento? perchè qualche volta mi trovo in difficoltà a riportarmi in una forma "comoda"

Re: Simulazione scritto d'esame 4

Inviato: martedì 3 giugno 2014, 14:06
da GIMUSI

Re: Simulazione scritto d'esame 4

Inviato: giovedì 5 giugno 2014, 18:26
da Gabe
Nell'esercizio 2 punto a, ti torna che un sottospazio possa essere [tex]W=Span[(0,1,0),(1,1,0)][/tex] oppure [tex]W=span[(1,1,0),(0,1,-1)][/tex]?

Re: Simulazione scritto d'esame 4

Inviato: venerdì 6 giugno 2014, 10:35
da GIMUSI

Re: Simulazione scritto d'esame 4

Inviato: venerdì 6 giugno 2014, 11:18
da Gabe
In simulazione scritto d'esame 4, esercizio 2, punto a, il quale chiede di determinare un sottospazio di dimensione 2 sul quale la forma quadratica risulta definita positiva

Re: Simulazione scritto d'esame 4

Inviato: venerdì 6 giugno 2014, 14:15
da GIMUSI

Re: Simulazione scritto d'esame 4

Inviato: venerdì 6 giugno 2014, 15:16
da Gabe
da [tex]q(x,y,z)=y^2+2xy-6yz+xz[/tex], mi sono riportato alla forma: [tex]q(x,y,z)=(x+y+1/2z)^2-x^2-7yz-1/4z^2[/tex].

A questo punto ho notato che se prendo [tex]x=0, y=t, z=0[/tex], quindi [tex]V1=(0,t,0)[/tex] ottengo [tex]q(V1)>0[/tex].

Se prendo [tex]x=t, y=s, z=0[/tex], ottengo [tex]V2=(t,s,0)[/tex] e [tex]q(V2)>0[/tex].

Quindi un sottospazio di dimensione 2 sul quale la forma è definita positiva è [tex]V=Span[(0,1,0),(1,1,0)][/tex].

Allo stesso modo ho ragionato per trovare un altro sottospazio [tex]W=Span[(1,1,0),(0,1,-1)][/tex].

Sinceramente la lineare indipendenza non l'avevo controllata, è un errore?

Re: Simulazione scritto d'esame 4

Inviato: venerdì 6 giugno 2014, 15:37
da GIMUSI

Re: Simulazione scritto d'esame 4

Inviato: venerdì 6 giugno 2014, 16:05
da Gabe

Re: Simulazione scritto d'esame 4

Inviato: venerdì 6 giugno 2014, 19:00
da Gabe
Avrei anche un altra domanda: Se la matrice associata alla forma quadratica B, la quale è simmetrica, fosse stata definita positiva, avrei potuto trovare gli autovettori ortogonali e avrei potuto normalizzarli...ottenuta la base ortonormale M, mi trovo una nuova matrice C diagonale, con la quale posso scomporre la forma quadratica, giusto?
Ora, visto che per a=1 la matrice non è definita positiva, non posso scomporre la forma quadratica mediante una nuova matrice C.
Però, posso comunque provare a diagonalizzarla, se autovalori e molteplicità me lo consentono, e scomporre la forma con questa nuova matrice diagonale D?

Re: Simulazione scritto d'esame 4

Inviato: venerdì 6 giugno 2014, 20:32
da Massimo Gobbino

Re: Simulazione scritto d'esame 4

Inviato: venerdì 6 giugno 2014, 21:29
da GIMUSI

Re: Simulazione scritto d'esame 4

Inviato: venerdì 6 giugno 2014, 21:41
da GIMUSI