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3. Consider the vector space V = R2[x] with its standard ordered basisE = 1,x,x2 and the linear map T :R2[x]−→R2[x], T(p...

3. Consider the vector space V = R2[x] with its standard ordered basisE = 1,x,x2

and the linear map

T :R2[x]−→R2[x], T(p)=p(x−1)−p(0)x2

  1. (a) (1 point) What is [T]E?

  2. (b) (1 point) Is T invertible?

  3. (c) (6 points) Compute the eigenvalues of T and their algebraic multiplicity.

  4. (d) (2 points) Is T diagonalisable? If so, find a matrix Q such that Q−1[T]EQ is diagonal. If not, findQ, so that the above matrix is upper triangular.

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Answer #1

V R T R TP)= -Bas PCt-1) Plo)x 2 No w T) - I - 1x 2 = -1 = O. 2= - 1 .2 Since detTJ) : Tu Invertnl Matix triangulur SInce eea

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