Let vectors ~ v 1 , ~ v 2 , ~ v 3 denote the columns of E = 1 2 1 1 3 5 1 2 4 Le

Question

Let vectors ~v1,~v2,~v3 denote the columns of

E =

1 2 1

1 3 5

1 2 4

Let R3 of E denote the vector space with ordered basis {v1,v2,v3}.
Define vectors u1 := 1+x+x2, u2 := 1+2x, u3 := 1+2x+x2.
Let (P3 )F denote the vector space P3 with ordered basis F = {u1 , u2 , u3 }.

(1) Draw the diagram for all the relevant transitions needed for the change of basis from R3E to (P3)F .

(2) Find the transition matrix from R3E to (P3)F .

(3) How can you check your work for the previous question?

(4)Suppose p = 3v1 + 2v2 -v3 and q = 3v1 -3v2 + 2v3 . Write vectors p and q as linear combinations of the u1, u2, u3

Write short phrases to explain your work please.

Explanation / Answer

E= V1 V2 V3 1 2 1 1 3 5 1 2 4 U1=1+X+X^2=[1,1,1]' U2=1+2X = [1,2,0]' U3=1+2X+X^2=[1,2,1]' F= U1 U2 U3 1 1 1 1 2 2 1 0 1 1 , 2 , 3

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