Consider the following linear transformation T : P_2 rightarrow

vazelinahS 2020-11-10 Answered

Consider the following linear transformation T:P2P3, given by T(f)=3x2f'.

That is, take the first derivative and then multiply by 3x2

(a) Find the matrix for T with respect to the standard bases of Pn: that is, find [T]ϵϵ, where- ϵ=1,x,x2,xn

(b) Find N(T) and R(T). You can either work with polynomials or with their coordinate vectors with respect to the standard basis. Write the answers as spans of polynomials.

(c) Find the the matrix for T with respect to the alternate bases: [T]AB where A=x1,x,x2+1,B=x3,x,x2,1.

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Expert Answer

Cristiano Sears
Answered 2020-11-11 Author has 96 answers

Sulotion: Given that nebce that to the T:P2P3T(f)=3x2f a) B={1,x,x2},γ={1,x,x2,x3} 

T(1)=0.1+0.x+0.x2+0.x3 

T(x)=3x21=0.1+0x+3x2+0x3 

T(x2)=3x22x=6x3=0.1+0.x+0.x2+0x3 

[T]Bγ=A=[000000030006]

b. That to that the N(1)Ax=0 [000000030006][x1x2x3]=[000] 3x2=0; 6x3=0; x2=0

Let then the n(1)=5pon|{1,0,x} spon let P(x)=7+8x+γx2P2 hence that to

Let then to T(P)=A[xBγ]=[000000030006][x0Bγ]=[003B6γ] 

Then the R(T)={(0,0,1,0)T(0,0,0)T} is spon {x2,x} c. A={x1,x,x2+1}B={x3,x,x2} Let the then T(x1)=3x21=0.x3+0.x+3x2+0.1
T(x)=3x2(1)=0.x3+0.1
T(x2+1)=3x2(2x)=6.x3+0.x+0.x2+0.1
[T]AB=[006000330000]

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