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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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 (c1−2c3)+(c2−12c4)t+4t2c3+8t3c4=0+0t+0t2+0t3 8t3c4=0t3⇒c4=0 4t2c3=0t2⇒c3=0 (c1−2c3)=0⇒c1−2(0)=0⇒c1=0 (c2−12c4)t=0t⇒c2−12(0)=0⇒c2=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 (c1−2c3)+(c2−12c4)t+4t2c3+8t3c4=−1+32t+8t2+15t3 8t3c4=15t3⇒c4=158

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}

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