Let T: P^3[- 1, 1] rightarrow P^3 via T(p) = p\' + xp Find the matrix representa

Question

Let T: P^3[- 1, 1] rightarrow P^3 via T(p) = p' + xp Find the matrix representation A for T, using the standard ordered bases B_ ab = {1, x, x^2} for P^2, and B_ = (1, x, x^2, x^3} for P^3 What are the dimensions of A? Find the Nullspace(A), Nullspace(T), Range(A), and the Range(T). Give both an implicit and explicit description of both Ranges. Give a basis for each subspace above, if it has one, and what is the dimension of each? Is T invertible? Why or why not? Is q(x) = 1 + 3x + x^2 + 2x^3 in the Range(T) ? If so, solve the equation T(p) = 1 + 3x + x^2 + 2x^3 for p(x). is q(x) = 1 + 3x + x^2 + 2x^3 in the Range(T) ? If so, solve the equation T(p) = 1 + 3x + x^2 + 2x^3 for p(x)

Explanation / Answer

Ans-

Here is a polynomial p(x,y)=(ax+by)2p(x,y)=(ax+by)2, it can be written like this

p(x,y)=([ab][xy])2p(x,y)=([ab][xy])2

and I know that it can also be written as something like vTMvvTMv, here v=[x,y]Tv=[x,y]T, and

M=[a2ababb2]

[ab][xy]=[xy][ab][ab][xy]=[xy][ab]

Then M=[ab][ab]M=[ab][ab]

Letting v=[xy]v=[xy] and A=[ab]A=[ab] we use the first line to note that Av=vTATAv=vTAT., and hence that

(Av)2=(Av)T(Av)=vTATAv=vT(ATA)v(Av)2=(Av)T(Av)=vTATAv=vT(ATA)v

So M=ATAM=ATA.

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