Define T: P₂ → R by T[p(r)] = the sum of all the coefficients of p(x). Problem 5. a. Use the dimension theorem to show that dim(ker 7) = n. b. Conclude that {x − 1, x² – 1, ..., x -1} is a basis of ker T. -

Q: 3. Let p₁(x) = ½x(x − 1), p2(x) = −x² + 1 and p3(x) = ½x(x+1). - (a) Show that E = {P1, P2, P3} is a…

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Q: 7. Prove that p, (t) = 1+t², p2(t) = t – 3t2, p3(t) = 1 +t – 3t, is a basis for P2.

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Q: { be a basis for R? . Find the vector æ if [æ]R =| 3 Let B 8 x = x = 6. x = x = H 00

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Q: Problem 4. T: M22 [a b] =[a+b C and (ii) find a basis of im T. You may assume that T is linear. M22;…

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Q: find the matna X for T elative to the basis B: T:R'R*, TG j) = (x-J,y-ta), 8'-{(1, -), CO 3)} I D

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Q: f(x) = 3 X. Let {(1,2) , (–2, –5)}, {{1, –2) , (–1,3)}, B C %3D be two different bases for R. a.…

A: As per Bartleby Instructions only first three can solved. Upload others separately. a. Let f be a…

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Q: #16. X=\ [5-24 dis are real numbers Then a basis for X 1S'. iS. A) 6) -2 3 2 3 D) E) Mone

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Q: Let T: R3 →R3 be a linear operator and B = {v1, v2, V3} a basis for R3. Suppose - 5,T(v3) Lo] T(v1)…

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Q: 5. Consider the basis aj = (1, 2, 3), az = (2, 4, 5), and az = (3, 5, 6) for R] × 3. Consider t E…

A: Given α1 =1, 2, 3, α2 =2, 4, 5, α3 =3, 5, 6 is the basis of R1×3 So let α = α1, α2, α3 Let τ = α…

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Q: Find

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