The set B = {1– 2t + t2,3 – 5t + 4t² , 2t + 3t²} is a basis of P2. Suppose C = {c1, C2, C3} is another basis of P2 such [4 11 that P = C+B Determine the polynomials c1(t), c2(t), and c3(t). 1 3 2 2 -1

Q: Which of the following sets of polynomials form a basis for P₂? (i) P₁ = 1 + 4x + 6x², P2 3+x = (ii)…

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Q: The set B=(3+2x², 12+2x+8x²,- (29+6x+20x2)) is a basis for P2. Find the coordinates of p(x) = 8 + 4x…

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Q: The first four Laguerre polynomials are 1, 1 –t,2 – 4t + t², and 6 – 18t + 9t² – t3. Show that these…

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Q: A basis B of R² such that e1B and e1 + e2 B are orthogonal, where e1, e2 are the standard

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Q: Determine whether S is a basis for P3. S= {3 – 4t2 + t, -2 + t², 2t + t³, 6t} O S is a basis of P3.…

A: S=3-4t2+t3, -2+t2, 2t+t3,6t Consider , a3-4t2+t3+b-2+t2+c2t+t3+6td=0 ; a,b,c,d∈ℝ…

Q: +t, 2t +t, 6t} O sis a basis of P O sis not a basis of P3.

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Q: 3) If V = R? and neither a, c nor b, c is a basis of R2, show that c 0. %3D

A: 3. It is given that a, c  is not basis of V ⇒c=λa    it is also given b,c is not basis of V⇒c=μb…

Q: Show that S = {t+1,t-1,(t-1)2} be a basis of P2, the vector space of all second-degree real…

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Q: Determine if S is a basis of P3. S = {1-t, 2t + 3t2, t2 - 2t3, 2+t3}

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Q: Let S = {x; 2 – x; x² – 2x}. A. S is a basis of P2 B. S is a basis of P C. None of the above

A: Let S=x, 2-x, x2-2xSince, we know that If V is a vector space of dimension n  (say) and…

Q: The set B = {2 – 4x?, 8+ 3x – 16x2, 21 + 9x – 40x²} is a basis for P2. Find the coordinates of p(x)…

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Q: Q1. Show that the first four Laguerre polynomials [named for the French mathematician Edmond…

A: The first four Laguerre polynomials are 1, 1 - x, 2 - 4x + x², 6 - 18x + 9x² - x³.

Q: Define T:R?→R? by T(x) = Ax. Find a basis B for R with the property that [T]B is diagonal. 2 - 4 A =…

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Q: Let A E Mpxn(F), and let B be the set of rows of A. Prove that B is a basis of F" if and only if…

A: The proof of given question is solved below.

Q: The first four Laguerre polynomials are 1, 1- 1,2-4t +t2, and 6 18t + 9t2-1. Show that these…

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Q: The set B = {4 – 2x2, 20 + x – 10x2, 28x2 – (55 + 3x)} is a basis for P2. Find the coordinates of…

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Q: Consider P2(R) with its basis B = (x, x + 1, x² –- 1). Find [x2 + 3x + 5]B.

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Q: Find a basis for the subspace 96 4a + 6b W = E R2x2 a, b eR} of R2x2- 86 5a Ex: 1 4 6. A basis for W…

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Q: Which of the following sets of polynomials form a basis for P2?

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Q: The set B = {3x² – 4, 152² – (20 + 2x), 38 + 4x – 27x²} is a basis for P. Find the coordinates of…

A: Given, the set B=3x2-4,15x2-20+2x,38+4x-27x2 is a basis for P2. We have to find the coordinates…

Q: B = (f1, f2, f3) is a basis for R2[] where fi = 1+ 2x + a², f2 = 2+ 3r + 2x², f3 = 2 + x +3x². If…

A: The correct option is option (e) 1 + 2x²   The detailed solution is as follows below:

Q: The set B = {2 + x², 6 + x + 3x?, 19 + 2x + 9x²} is a basis for P2. Find the coordinates of p(x) (39…

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Q: Let T : P2(R) – P2 (IR) be defined by T(p(x)) = p(x) – 3p'(x), and let ß denote the standard ordered…

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Q: Let B be the basis of P, consisting of the three Laguerre polynomials 1, 1-t, and 2-4t+t, and let…

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Q: Let B = {x²+x,-2x² + 3x + 3, 2x² – x – 2} be a basis of P2(R)

A: Given,            B=x2+x,  -2x2+3x+3,  2x2-x-2

Q: Determine whether S is a basis for P3. S = {5 – t, t³, 4t2, 2t + t³, =1 + 5t} O is a basis of P3. O…

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Q: Let V = P4(R). Find a basis of V containing the polynomials p1(x) = 1 + 2x-x 2+3x 3+2x 4 , p2(x) =…

A: The detailed solution of the problem is given in the next steps.

Q: Use the standard basis for P2, B = {1, t, t²}, and the coordinate mapping to show that 10 – 12t –…

A: The solution is below.

Q: Determine whether S is a basis for P3. S = {5 – 3t2 + t3, -2 + t2, 3t + t³, 5t} O is a basis of P3.…

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Q: 6. Find a polynomial p(x) for which {r² – 1, x² +x +2, p(x)} is a basis of P2(R).

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Q: Determine whether S is a basis for P3. S = {4 – 2t2 + t, -4 + t2, 3t + t2, 6t} O is a basis of P3. O…

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Q: 2. Let B₁ be the basis r²e, ze, e of V = span(r²e², xe, x), and B₂ be the basis is H.6 of w that the…

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Q: Suppose v1, v2, v3, V4 is a basis of V. Prove that vi + v2, V2 + V3, V3 + V4, V4 is also a basis of…

A: we are given that v1,v2,v3,v4 is basis of V. we have to prove that v1+v2,v2+v3,v3+v4 ,v4  is also…

Q: Determine whether S is a basis for P3. S = {3 – 2t2 + t3, -2 + t², 3t + t³, 5t} S is a basis of P3.…

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Q: Lemma 2.8 B is a basis of T iff

A: To prove that Β is a basis of Τ iff ∀Ο∈Τ ∃ Β0⊂Β: Ο=∪ ΒΟ

Q: Let B = {1+x, 1– 3x?, x + 6x²} and p(x) = 12 + 8x + 9x?. You are given that B is a basis for P2. If…

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Q: Let B={1, x, x², x³} be a basis for P3, and let P={P1, P2, P3, P4} be the set of polynomials given…

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Q: Let B = {x – 2, x² - 2,4x}. %3D Which of the following statements is true? B is a basis of P3 B is a…

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Q: Determine whether S is a basis for P3. S = {2 – t, t³, 7², 8t + t³, -1 + 2t} S is a basis of P3. S…

A: It is to be checked whether S is a basis of P3 or not where  S=2-t, t3, 7t2, 8t+t3, -1+2t

Q: Consider B find [p(x)]B. {x2 + x, 2 + 2x, 1} an ordered basis of P2(R). If p(x) = 4.x² 2x – 6, %3D

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Q: Let S = {x; 2 – x; x² – 2x}. A. S is a basis of P2 B. S is a basis of Pa C. None of the above

A: The answer is:  (A) S is a basis of P2. Proved below.

Q: Let {f1, f2, f3} be the dual basis of {e1, €2, €3} for R³, where e₁ = (1, 1, 1), €₂=(1,1,−1), Find…

A: Given basis are e1,e2,e3 of R3 where e1=1,1,1e2=1,1,-1e3=1,-1,-1 Its dual basis are f1,f2,f3 We have…

Q: 60. In R4, given a nullspace V = {(x1; X2; X3; X4) E R4|¤1 – *2+ x3 = 0 A x2 + x3 + x4 = 0}. %3D…

A: see below the answer and explanation

Q: Let B = {x – 2, x² – 2,4x}. Which of the following statements is true? B is a basis of P3 B is a…

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Q: B = (f1, f2; f3) is a basis for R2[r] where fi = 1+z+2r², f2 = 1+2r + 5x², f3 = 2+1+2r². If g e…

A: Coefficient of linear combination is given. We have to find the element

Q: Let 0 = V5 and K = Q(0). Two bases for K over Q are a = {1+0,1+ 0²,1+ 0³, 1 + 0ª} and B = 1+ 0²…

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Q: Let P2 be the polynomials of degree no more than 2. Is {x2+x+1,2x+1,3x2+1} a basis for P2 ?

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Q: The set B = {x² – 2, 4x² − (8+3x), 15x² − (28 + 9x)} is a basis for P₂. Find the coordinates of p(x)…

A: Result: Let B be the basis of P2 and let p(x)∈P2. Then the coordinates of p(x) relative to the basis…

Q: Determine whether S is a basis for P S = {5 - 4t2 + t, -4 + t2, 2t + t, 5t} O is a basis of P2. O s…

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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.