2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider B = {1+x = 2a°, –1+ x + a*, 1 – x + a²} , B' = {1- 3x + a², 1 – 3x – 20², 1 – 2x + 3a²}. (a) Show that B and B' are bases of P2(R). (b) Find the coordinate matrices of p(r) = 9x² + 4x – 2 relative to the bases B and B'. (c) Find the transition matrix Pg→B'. (d) Verify that [x(p)]B = PB-¬B' [x(p)B].

Q: 6. Suppose V is a complex vector space with dimension n and T E L(V) is invertible. Let p and q be…

A: In the question it is asked to prove p(0)q(z)=znp(1z).

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A: Given: (a) If u is orthogonal to both v and w. To prove:  u is orthogonal to v + w. (b) If u is…

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A: Given force is F→x,y,z=x2,4xy3,xy2 Let r→=rt→ be the boundary of the rectangle with vertices  0,0,0,…

Q: Let P2 be the vector space of all real polynomials of degree at most 2. Let p1, P2, P3 E P2 be given…

A:

Q: 2b) Let P(x) = P2x² + P1x + po and ¶(x) = q2x² + q1ª + q0 polynomials from R=ls2•. Let be given the…

A:

Q: 27. Let S be the set of all vectors both of y in R' such that x + y -z = 0 and 2x – 3y + 2z = 0.…

A:

Q: Let v= [x-2y, x+2y², z] be a vector field. 1- Compute Curl(v). 2- Find the condition on x, y, and z…

A: From the given problem:   v=x2-2y2,x2+2y2,z   It means:   vx=x2-2y2vy=x2+2y2vz=z   So,   a).…

Q: Let V be the subspace of the vector space of all real-valued functions that is spanned by the setS…

A: If V is the subspace of the vector space of all real-valued functions that is spanned by the set S…

Q: 41.) Use given grid to sketch the vector field F = (y, -x). Include vectors at all integers points…

A: Given vector field is F=12y,-x

Q: For a polynomial p ao + a₁x + a₂x² + + anx, let p denote the polynomial obtained from p by removing…

A: Note:  As you requested only Part C(I) so I answered here only that.

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A:

Q: Find the divergence of the vector field. F(x, y, z)= 6x¹0i-10xy ¹0j O divF(x, y, z) = 60x⁹-100xy⁹ O…

A: To find the divergence of the given vector field.

Q: a. The zero vector 0 is in ker(A) b. If x and y are in ker(A), then so is z = x + y. c. If x is in…

A:

Q: Consider the vector space P, of all polynomials of degree = a, a,+b, b2 , then the inner-product of…

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Q: 2. a) Let V == R[r]<n be the vector space of real polynomials of degree at most n. Define a function…

A: Axioms of Euclidean space: Let R be a vector space, let V be a vector space over R. V×V→R that is…

Q: (a) Show that the vector field F (x, y, z) = (x² + y², x, z³) is not conservative.

A:

Q: Given the 2-D vector field : G(x, y) = (2y)î + (x)j and the triangular curve T : line segments from…

A: Given: G→x,y=2yi^+xj^ Triangular curve T, line segments from point A0,0 to point B4,0 to point D4,2…

Q: Find the divergence of the vector field. F(x, y, z)= 7x¹i, 7xy7j O divF(x, y, z) = 7x³¡— Ţxy³j…

A:

Q: 2. Show that any vector field of the form F(r, y, z) = f(y, z)i + g(x, z)j+ h(x, y)k is…

A:

Q: b) Find the curl of the vector field, CurlF, if F= Axzi- By 2j+ Cx tk, %3D where the coefficients A,…

A:

Q: 1. Find the divergence and curl of the following vector fields a. F = (1/y²)j b. F=y³z³i+ 2xyz³j +…

A: We have to find the divergence and curl of the given vector fields.

Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B' A) Given the…

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Q: Use the inner product (f,9) = f(z)g(x) dz in the vector space C"0, 1] of continuous functions on the…

A: It is given that fx=-5x2-3 and gx=9x-7. We have to evaluate f, g, f, g and αf, g where αf, g is the…

Q: Consider the space of square integrable functions L² ([–1, 1]). Let B = {Pn|n = 0, 1, 2, ...} the…

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Q: Consider the vector space P,(1)of the polynomials of degree less than or equal to 2. Then check…

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Q: Let B = {1, x, x², x³, x4}, which is a basis for the vector space of polyno- mials W = P4(R). Let V…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: Consider the scalar field f(x,y, z) = 2x² – y² + 3z2 i. V Vfl(1.-1.1) Determine II. VxVfl1.-11)

A:

Q: Let v = [x2-2y², x²+2y², z] be a vector field. %3D 1-Compute Curl(v). 2- Find the condition on x, y,…

A:

Q: b) Find the curl of the vector field, CurlF, if F = Ax²z²i– By:'j+Cz*k, where the coefficients A,…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Question 2 (a) Verify that P(R) with addition and scalar multiplication %3D defined by Eaxe£bx' =…

A:

Q: 45. Let {v, w} be an orthonormal basis for R², and let T: R2 → R² be the function defined by T(u) =…

A:

Q: Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the inner-product of…

A: We have to find the inner product of the following -  p = 2 + 1.7x q = 4.8 + 3.2x

Q: 5. Let V be a vector space, and let x1,X2, ... , Xk E V. 5(b) Prove that if x1, X2, ..., Xk is…

A:

Q: 2x 2y Consider the vector field F = |x² + y2’ x² + y² (a) If C1 is the unit circle x² + y² = 1,…

A: Compute the circulation and flux using parameterization:  ∮C1Fdr=∮C12xx2+y2,2yx2+y2·dx,dy…

Q: Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+x – 2x?, –1 +x+x², 1…

A: Consider the provided question, Given that,  B=1+x-2x2, -1+x+x2, 1-x+x2B'=1-3x+x2, 1-3x-2x2,…

Q: In the vector space of polynomials of degree at most two, coordinates of the vector p()=5+x+5x2 in…

A: Given p=5+x+5x2and Basis B={e1 =1+x, e2=x+x2 ,e3=2+x}

Q: 10. Consider the plane 2x - 3y+4z 0. Find a basis B of this plane such that (B = for ū =

A: Solution:-

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A: Part 1. (x, y) is in the first quadrant.  =&gt; x &gt; 0 and y &gt; 0 =&gt; x2 &gt; 0 and - y &lt; 0…

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Q: a) From the vector space of polynomials of degree at most 3 to itself the following linear…

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Q: 3. Let V be the set of column vectors of the form V = : x, y, z E R, x > 0 with vector addition…

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Q: Show that V x VG = 0 for all scalar field G(x,y,z). %3D

A:

Q: a) T: M, → R, where M, is the vector space of n by n matrices over a field, and T (A) = det(A). b)…

A:

Q: Q3. Let L: R2R2 be a mapping from the vector space R2 to itself defined by L(x , y)= ( 2x+y, x-3y)…

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Q: /- X be the vector Space of all Complex 2x2 and de fine T: x →x by Tx = bx, where Let matrices bEX…

A: Given :-  Let X be the vector space of all complex 2×2 matrices and define T : X →X  by Tx = bx ,…

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+x - 2a?, -1 +x +…

A: Note: According to bartleby we have to solve only first question please upload the question…

Q: a) T: M, → R, where M,n is the vector space of n by n matrices over a field, and T (A) = det(A). b)…

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Q: , Find a basis {p(x), q(x)} for the vector space { f(x) e P3[x] | f'(-5) = f(1)} where P3[x] is the…

A:

Q: Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the magnitude of…

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Q: If a vector field A(x,y,z)=xz3ex - 2x2yzey+2yz4ez , calculate the divergence ∇·A at point…

A: A vector field is given as  and

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.