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 R2[r] and [g]s = -2 then g = a) 1– 2r + 3r² b) 5z – 2r2 c) 5 – 2r² d) 5+1– 2r² e) 4 +x – 2r2

Q: Find an orthonormal basis of ℋ:  x− 2y − 3z = 0.

A: Introduction: The Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner…

Q: Find a basis for the plane 4 + 3y + 5z = 0 in R. %3D

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Q: the coordinates of (2, 3, 4, – 1) with B = {(1, 1, 1, 2), (1, –1, 0, 0), (0, 0, 1, 1), (0, 1, 0, Q}}…

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Q: Show that {u,, u2, uz) is an orthogonal basis for R. Then express x as a linear combination of the…

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Q: 18. Use the Gram-Schmidt process to transform the given basis (w, w, orthogonal basis where w =…

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Q: In the vector space P,, of polynomials of degree at most two, the matrix of transition from the…

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Q: Explain why S i not a basis for P2. S = {1, 4x, -5 + x?, 3x} O sis linearly dependent. O s does not…

A: We have to find out

Q: The vectors u, v, w have the coordinates (-6, 2, 1), (2. -1,0) and (7. -2, -1) respectively relative…

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Q: Find a basis for the set of vectors in ℝ2 on the line y = 5x.

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Q: Let S = {v1, v2} be a basis of IR? where vi = and What element of R? is determined by the linear V2…

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Q: 1. Consider the basis B = {(2, –v2), (2, V2)} for R². Define (ỷ, 2) =y,2 +y22. %3D a) Verify that B…

A: Note:- As per our guidelines, We can answer the first question since the exact one wasn’t specified.…

Q: ,T If {v1, ..., Vn} is an orthonormal basis for R", then vịv¡ + · · ·+ vn Vn = I. True False

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Q: Explain how fixing a basis v1, corresponding function in n variables of the form q(x1, · ·· , Xn) =…

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Q: Find the matrix for T relative to the basis B, and use Theorem 8.5.2 to compute the matrix for T…

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Q: The Gram-Schmidt process transforms the basis u,=(1, 3), u,=(1, –7) of R² into the orthogonal basis…

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Q: 3) Let C be a basis that contains vectors (1 + (-1) x + 1x² + 1x³), (1 + 2x + 2x² + (−1) x³), (1 +…

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Q: 3) Let S contain vectors of the form f(x) = a + bx + cx² + dx³ : f(-2) = 0 and f'(-4)= 0: Then a…

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Q: -3' 2, uz [3] -4 is an orthogonal basis of R3 and x = 2 Let u1 1, u3 4 If x can be expressed as a…

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Q: the basis B=(,=4-x?,e,=-2+3x, e,=1+x} to the basis A= {a,=1, a,=x, a,=x} is: O-3/10 In the vector…

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Q: Let TilP > Pz, the basis of NIT) T (ax²+bx+c) = 2Qx+ b. Then %3D is

A: Applying nullity definition we have

Q: Take L: V --> W, from P2 --> P2 and L(p) = p + p', with basis B = {1 + x + x2, 1 + x, 2x + x2} and…

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Q: (10) R¹3x+2y-20). Determine an orthonormal basis of W

A: Given : W={x,y,z∈R3:3x+2y-z=0}

Q: 1. Find [p]a, relative to the basis 3 = {P1. P2: Ps} for P, where p = 4- 3x +x², P1 = 1, P2 = X, P3…

A: Given

Q: The Gram-Schimidt process transforms the basis u=(1, 3), u2-(1, -7) of R intothe orthogonal basis…

A: Given u1=⟨1,3⟩,u2=⟨1,−7⟩

Q: Show that {e1,€2, - } is a Schauder basis for P, 1<p< ∞, where S1 ifi=n (E") en = %3D 0ifi#n.

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Q: Find the matrix A' for T relative to the basis B'. T: R2 - R2, T(x, y) - (-5x + y, 5x - y), B'- ((1,…

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Q: now that B = {x, r³ + 2x² + 4, x² + 2, 1} is a basis for P3.

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Q: Suppose that you have three vectors: f1 (x) = 1, f2 (x) = x – 1, and f3 (x) = ; (x² – 4x + 2), that…

A: Suppose that you have three vectors: f1(x)=1, f2(x)=x-1, and f3(x)=12x2-4x+2, that make up an…

Q: Define a basis v1, v2 of R² by vị obtained by applying the Gram-Schmidt Procedure to v1, v2? 2 by…

A: Consider the provided question, A Basis of a vector space. Let V be a subspace of Rn for some n. A…

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: Consider an inner product on two P3 vectors as (f, g) = f(-1)g(-1) + f(0)g(0) + f(1)g(1) + f(2)g(2):…

A: Since you have posted multiple subpart question, we will solve the first three sub parts for you. If…

Q: Show that B = {v1, v2, v3} is an orthogonal basis for R3: v1 = (3,−3,0), v2 = (2,2,−1), v3 =…

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Q: Let s = {(r² –3r+4, 2r-5, x²+1} be a basis for P2, let q E P2. If (q). = [2,-3, 1], then q = %3D %3D…

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Q: Consider the basis B= {1, x+x2, 2x2-1} for the vector space P,. The coordinate vector of the…

A: Given that the basis B={1,x+x2,2x2-1} for the vector space P2. Here we have to find the coordinate…

Q: Let V = span {1+ 2x – x² + 3r³, -2+x+x² – x*, -1+ 8r – x² + 7r", 3 + x – 2r² + r'}. Find a basis ß…

A: Step:-1 Note:- To span V we need the polynomials of degree 3, so dimension of V is 4. (think using…

Q: Show that the set S = {p1.P2-P3} is a basis for P2 (space of polynomials of degree less or equal to…

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Q: Let T: R3 → R³, T(x, y, z) = (2x + 2y – 5z, 2x – 5y + 2z, -5x + 2y + 2z) be an endomorphism. Is T…

A: The given linear transformation is T(x,y,z)=(2x+2y-5z, 2x-5y+2z, -5x+2y+2z) Considering the standard…

Q: Find the matrix for T relative to the basis B, and use Theorem 8.5.2 to compute the matrix for T…

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Q: Explain why S is not a basis for R2. S - {(-5, 2)} O sis linearly dependent. O s does not span R2. O…

A: Basis :  The set of vectors v1,v2,...,vn ⊆V in a vector space V is called a basis for V if (i) S is…

Q: If {v1,..., Vn} is an orthonormal basis for R", then vịvi + · · ·+ vnvn = I. O True O False

A: Given, v1, v2, ..., vn is an orthogonal basis for Rn. then v1v1T+...+vnvnT=I It is True or False. To…

Q: Let {u₁, U2, u2} be an orthonormal basis for an inner product space V. Suppose v = au₁ + bu₂ + cuz…

A: We have v=au1+bu2 +cu3 Now, ∥v∥=a2+b2+c2 Given ∥v∥=80 So,  80=a2+b2+c2⇒802=a2+b2+c2⇒80=a2+b2+c2…

Q: Find a basis for the plane 4x + 3y + 5z = 0 in R³.

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Q: 6. Define fe (R2)* by f(x, y) 2x + y and T: R2 (3x + 2y, x). (a) Compute T'(f). (b) Compute [T']p,…

A: We will use the basic knowledge of linear transformations to answer all the three given subparts of…

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

A: Given that S=1-x,1-x2,-1-6x+7x2 Explain why S is not a basis for P2 if S is linearly dependent the S…

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: If x, x is a basis of solutions of r'y" + ary'+ by = 0, %3D then a = (A) -2 (B) 2 (C) -3 (D) 3 (E)…

A: Just use definition of solution of differential equations.

Q: suppose that x↓1,..,x↓(n-1) are linearly independent vectors in R^n and also x↓n is notin the span…

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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: 10)Define TE L(R²) by T(x, y) = (41x + 7y,-20x + 74y). Verify that the basis of T %3D with respect…

A: Question 10. The detailed solution is as follows below: