Consider the following linear transformation T:P2→P3, 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=x−1,x,x2+1,B=x3,x,x2,1.

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Consider the following linear transformation T:P2→P3, given by T(f)=3x2f&#039;.  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=x−1,x,x2+1,B=x3,x,x2,1.

Cristiano Sears · Accepted Answer

Sulotion: Given that nebce that to the T:P2→P3T(f)=3x2f′ a) B={1,x,x2},γ={1,x,x2,x3} T(1)=0.1+0.x+0.x2+0.x3 T(x)=3x2⋅1=0.1+0⋅x+3x2+0⋅x3 T(x2)=3x2⋅2x=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=0Let then the n(1)=5pon|{1,0,x} spon let P(x)=7+8x+γx2∈P2 hence that toLet 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={x−1,x,x2+1}B={x3,x,x2} Let the then T(x−1)=3x2−1=0.x3+0.x+3x2+0.1T(x)=3x2−(1)=0.x3+0.1T(x2+1)=3x2(2x)=6.x3+0.x+0.x2+0.1[T]AB=[006000330000]

Consider the following linear transformation T : P_2 rightarrow

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