7. Consider the ordered bases B = (sin(3x), cos(3x), sin (3x), cos (3x), sin(3x) cos(3x)) and C = (1. cos(6z). sin(6z), cos(3z), sin(3z)) for a subspace W of C (R). (a) Find the transition matrix from B-coordinates to C-coordinates. (Hint: While this problem can be done using row reduction, it is easier, in- stead, to consider various trig identities that you learned in calculus.) (b) Use your answer to part (a) to express f(x) = sin(3x) (1 - 2 sin(3x) + 5 cos(3x)) in C-coordinates.

Q: Given the bases B = {(1,-1), (2,0)} and B' = {(1,2), (0, -1)} of R2 find: a. The transition matrix…

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Q: (a) Find the Jacobian matrix ду əx where y -B I2 I3 3+11 2+2r2 and x = 1+373

A: (a) Given y=3+x12+2x21+3x3 and x=x1x2x3   To Find: The Jacobian Matrix ∂y∂x

Q: Ke Find the matrix M of the linear transformation T : R → R° given by - (E) I -6x1 + x2 + 7x3 -6x1 +…

A: Solution

Q: Consider f(t) = (cos t, sin t, t?) and g(x, y, z) = (x² + y² + z², 3x + 4y – 12z) and h = g(f(t)) =…

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Q: Given the bases B = { f1= sin x, f2= cos x } and B' = { g1= 2sinx+cosx, g2 = 3 cos x } for a…

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Q: (f) What is the first-order condition implied by unconstrained maximization of I1 X 1333 X, where x…

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Q: Find the transition matrix from B to B'. B = {(0, -1, -2), (0, 4, 1), (1, 3, -3)}, B' = {(1, 0, 0),…

A: The objective is to find the transition matrix from B to B'. Write B in terms of B' to find the…

Q: T(x, y) = (-x, 4x + 8y, 9x − 3y, 6y – 5x). Find its associated matrix A. A = JUL

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Q: Let g: R- R$ AND f: R3→R² be a c² function. g(5)=[3].f(1,3,2)=(3,0), g'(5)=(-2,2,2), and 2 -2 -1…

A: Given the functions g:ℝ→ℝ3 with g5=132, g'5=-222, and f:ℝ3→ℝ2 with f132=30, Df132=-2-1-225-2. We…

Q: 8. We have seen that the matrix R(0) = ( cos sin 0 around the y-axis rotates vectors in R2…

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Q: Consider the subspace W of D, given by W = span(e2x, e2x cos x, e2x sin x).       (a) Find the…

A: The given subspace is W=Spane2x,e2xcosx,e2xsinx. (a) The basis is B=e2x,e2xcosx,e2xsinx, To Find:…

Q: The function f : R³ → R' is linear and f(1,0, 0)) f(0, 1,0)) f(0,0, 1)) -1 5. (a) Find the matrix A.…

A: We will use the fact that the columns of A are images of basis vectors of the domain under f.

Q: a. y" +0.5y'+2y = 0 b. 2y" +0.5y+ 8y = 6 sin(2t) c. y" +0.25y + 4y = 2 cos(3t), y(0) = 1, y'(0) = –2…

A: In this question, we have several differentials equations are given. By reducing the order of…

Q: y = [ 0 1] (2x2 matrix) + [-1 0] d/dx y [1] (1x2 matrix) y (0) [1] [1] %3D [2]

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Q: (1x30'+ 3xD+1)y:о, усідів, у'(1)-о утобу'+0.09у= y (0) = 2.2, y' (0) = 0.14 -0.3x -o-3x e хе

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Q: Verify that the vector Xp is a particular solution of the given nonhomogeneous linear system. dx dt…

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Q: 3. If B= (-3, 2), (4, -2) and B' = (-1,2), (2,-2) are two bases for R² (a) Find the transition…

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Q: = {(1,2), (-1, –1)}, B' = {(-4, 1), (0, 2)} be bases for R². %3D 2 1 Let A = be the matrix for T :…

A: Given that B=1, 2, -1, -1, B'=-4, 1, 0, 2 be two bases for ℝ2. And A=210-1 be the matrix for T :…

Q: Give the General Solution.  (d/dt)(vector(x)) = matrix(2,3,-3,2)vector(x)

A: Given that,  ddtx1x2=23-32x1x2 The corresponding linear equations are, x1'=2x1+3x2x2'=-3x1+2x2…

Q: Use the definition of the matrix exponential to

A: Given,           X'(t)=01-10X(t)

Q: 1. Let A=M³₂ (f) be the matrix B' associated with the linear map f(x,y,z) = (3x − y +…

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Q: Let A = M. (f) be the matrix associated with the linear map ƒ((x,y,z)) = (x-2y,y+2,0) relative to…

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Q: I don’t understand here how they got the associated matrix like the part with f(1,0,0,0) and so on…

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Q: Exercise 3.1. Find f'(x), Vf(x) and V²f(x) for the following functions (a) f(x) = }r"Hx, where H is…

A: Given, fx=12xTHx, where H is an n×nconstant matrix.…

Q: Consider the functions f(x, y, z) = xyz and g(t) = (cos(27t), sin(27t), e*) 6. (a) Compute Df(1,0,…

A: f(x,y,z)=xyz(a) Df=i^∂f∂x+j^∂f∂y+k^∂f∂z   =i^∂∂x(xyz)+j^∂∂y(xyz)+k^∂∂z(xyz)…

Q: Find the gradient vector and the Hessian matrix for each of the functions following:

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Q: 4 3 3. Let XT = [X₁, X2, X3, X4] with mean vector μ = Partition X as X = /X₁ X₂ X3 X4 Let A = (1 2)…

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Q: T(S)-(S(1)." (1)) where P.(R) is the set of polynomials of degree 4 or less and f(1) is the…

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: (b) Find the Jacobian matrix ду əx where y [1+1+2+3 -x1x2x3 -[1+5+ = -B X2 I3 and x =

A: (b) Given y=1+x1+x2+x3-x1 x2 x3 and x=x1x2x3   To Find: The Jacobian Matrix ∂y∂x

Q: Find the transition matrix from B to B'. B = {(1,0) , (0, 1)}, B' = {(1, 1), (5, 6)} 6 -5 5 o [1 5 6…

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Q: 4. Consider the vector space P, and the bases S = {1,x,x²} S' = {3,x-1,x² - 2x}. Find the transition…

A: We can solve this as follows:

Q: Let B- ((1, 0, 1), (1, 1, 0), (0, 1, 1)) and 8- ((1, 0, 0), (0, 1, 0), (0, 0, 1)) be bases for a,…

A: Note : '' As per guidelines we will solve first three subparts. If you want any specific subpart to…

Q: 3. Express the matrix cos sin 0 B = sin 0 0 cos 0 as a successive proper and improper orthogonal…

A: Given: The matrix is B=cosθ-sinθ0sinθcosθ000-1. To find: The orthogonal transformation. The…