Let Bo = {1, X, X²} be the canonical base of R2[X] and put %3D 1+X+2X = 1+X+X2 = 1+2X +X² E1 !! E2 E3 Prove that B {€1,€2, €3} is a basis of R2[X]. Give the dual basis B of B.

Q: Determine whether S is a basis for R3 S={ (0, 3, -2), (4, 0, 3), (-8, 15, 16)} OS is a basis of R3…

A: Set S forms a basis of ℝn if S is Linearly independent. S spans ℝn.

Q: Determine whether S is a basis for the indicated vector space. S = {(0, 0, 0), (3, 6, 5), (5, 1, 3)}…

A: Given S = {(0, 0, 0), (3, 6, 5), (5, 1, 3)}

Q: Prove that a function T: Fn→Fm is linear if and only if there exist f1, f2, . . . , fm ∈ (Fn)∗ such…

A: To prove the function T: Fn →Fm  is linear if and only if there exist f1, f2, f3, f4 …..., fm.

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: -5a1 Let T : P1 → R² be the isomorphism given by T(a1x + a2) -2a2 Using the basis {5x – 8, –4x}for…

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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: Explain why S is not a basis for P2. S = {1 - x, 1 – x², -1 – 3x + 4x²} O is linearly dependent. O s…

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Q: Let B be the standard basis of P₁, and consider the basis B' [T]B¹,B. = {(1)· (9)}< of V. Find

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Q: Let B = be basis for R. Find the dual basis B* of B

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Q: Let ẞ= {v1, ••• , vn} be an orthonormal basis for Rn. (a) Prove that, for any x and y in Rn, x · y =…

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Q: Let B be the basis of R² consisting of the vectors and let C be the basis consisting of {B} Find a…

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Q: Let, a1, a2 and a3 are the basis for R. [R1] y = |R2 [R3] a1 = |0 a2 = |0 a3 = Find the following:…

A: Since you've asked multiple questions, as per our answering guidelines we are supposed to solve the…

Q: Let B be the basis of R² consisting of the vectors and let C be the basis consisting of Find a…

A: Solution:-

Q: 2. Let B₁ be the basis 2²e², re², e² of V = span(2²e², zeª, z), and B₂ be the basis of that the map…

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Q: Determine all values of k so that S= {k- kx2,3+ kx,2 + x + kx²} is a basis for P2, the vector space…

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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: Explain why S is not a basis for P2. S = {1, 7x, -6 + x², 2x} O sis linearly dependent. O s does not…

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Q: Consider the multiplication operator LA : R² → R defined by LA(x) = Ax where A = Find an ordered…

A: An ordered basis for a vector space V(F) of dimension n is a basis {u1,u2,...,un} together with a…

Q: -2 Let T : R? → R² be defined by Tx = Ax where A = Find a basis 10. B for R² such that the B-matrix…

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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: ii) Show that S = {(t – 1)³, (t – 1)², (t – 1), 1} is a basis of P3(t) where P3(t) stands for the…

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Q: .3 Let W = span{x, x' x³, x>} in C[0,1]. Use Gram Schmidt to find an orthogonal basis for W.

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A: (a).               Orthonormalize the set of the vectors                      v1 = 1111 , v2 =…

Q: (a) Prove that the set ẞ = {p0(x), p1 (x), ... , Pn(x)} of Lagrange polynomials is linearly…

A: Given: ẞ = {p0(x), p1 (x), ... , Pn(x)} To Prove" (a). Prove that the set ẞ of Lagrange polynomials…

Q: Let n >= 3 be an integer. Let Vn(x) = {[x 1 1 1 ... 1 1]T,[1 x 1 1 ... 1 1]T, [1 1 x 1 ... 1 1]T,…

A: By the consequences of vector spaces we solve the given problem.

Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x², –1 – 6x + 7x²} O s is linearly dependent. O…

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Q: Determine whether S is a basis for P3. S = {8 - t, t3, 7t2, 2t + t, -1 + 8t} Os is a basis of P3. O…

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Q: Explain why S is not a basis for P2. S = {1 - x, 1 – x4, –1 – 2x + 3x2} O S is linearly dependent. O…

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Q: 21: Let L: M22 → M22be defined by L(A): =E [1 2j [3 4,for AinM22. Consider the ordered basis or M22.…

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Q: (ii) Find the dual basis { V4. } f the stan dard basis of IR. Led T. in 3

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Q: Let A = {v1, vz, - also a basis for pV? How about the converse, i.e. if B is a basis for FV, is A…

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Q: Let {₁,..., wk, Wk+1,...,n} be a basis of R" and c₁,...,CER be nonzero scalars. (a) Prove that S =…

A: Note that we used one fact in this prove which state that if B is basis of V and C is any linearly…

Q: Exercise 4: For V = R³, define U = a +b+c= 0 }W = a - b+ 2c = 0 1. Find bases for U, W 2. Find a…

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Q: Find the orthogonal complement W⊥ of W and give a basis for W ⊥

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Q: Explain why S is not a basis for P2. S = {1 - x, 1 - x², -1 - 4x + 5x2} O sis linearly dependent. O…

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Q: Let P2 denote the vector space of all polynomials in the variable a of degree less than or equal to…

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Q: Consider P2, endowed with the following inner product: 1 1 (p, q) : G P(0)q(0) + G P(1)q(1) + for…

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Q: Let IF = R and W = {(#1, #2,... , tn) € R* | #1 + *2 + · . · + &n = 0}. (a) Let n :4, so that W =…

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Q: Explain why S is not a basis for P2. S = {1, 6x, –5 + x², 8x} O sis linearly dependent. S does not…

A: Basis: A non-empty subset B of a vector space is basis if a) B is linearly independent b) B spans V.…

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: -{圈[} B O B is an orthonormal basis for R?. O B is an orthogonal basis for IR?, but it is not…

A: Given B=1/21/2,1/2-1/2

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: Explain why S is not a basis for P2. S = {1, 8x, -6 + x², 3x} O sis linearly dependent. O s does not…

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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 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: Explain why S is not a basis for P₂. S = {1, 6x, -2 + x²,4x} O S is linearly dependent. OS does not…

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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.