Consider these Values: A=32, B=15 and C=17. The set ot first four Hermite polynomials is. S={1,2t,-2+4t^2,-12t+8t^3}

allhvasstH 2021-09-05 Answered

Consider these Values: A=32,B=15andC=17
The set ot first four Hermite polynomials is
S={1,2t,2+4t2,12t+8t3}. These polynomials have a wide variety of applications in physics and engineering.
1) Show that the set S of first four Hermite polynomials form a basis for P3
2) Find the coordinate vector of the polynomial p(t)=1+At+8t2+Bt3 relative to S

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Expert Answer

pivonie8
Answered 2021-09-06 Author has 91 answers

Given:
Consider the values A=32,B=15andC=17
The set of first four Hermite polynomials is S={1,2t,2+4t2,12t+8t3}
Calculation:
a) dim(P3)=4
The standard basis for P3={1,t,t2,t3}
For, c1(1)+c2(t)+c3(2+4t2)+c4(12t+8t3)=0
(c12c3)+(c212c4)t+4t2c3+8t3c4=0+0t+0t2+0t3
8t3c4=0t3c4=0
4t2c3=0t2c3=0
(c12c3)=0c12(0)=0c1=0
(c212c4)t=0tc212(0)=0c2=0
then c1=c2=c3=c4=0
Answer (a):
c1=c2=c3=c4=0
The polynomials are Linearly Independent and hence they form a basis for P3
b) P(t)=1+At+8t2+Bt3=1+32t+8t2+15t3
(c12c3)+(c212c4)t+4t2c3+8t3c4=1+32t+8t2+15t3
8t3c4=15t3c4=158

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