Suppose L: R → R° be defined by u, +17 Lu2 2u2 %3D Uz Uz Determine whether L is a linear operator or not.

Q: let ej denote the vector whose jth coordinate is 1 and whose other coordinates are 0. Prove that…

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Q: Consider the following linear operator; T : R → R° given by T(x, y, z) = (y, x + z, z).

A: We have to find the real Eigen value of T

Q: In Fn, let ej denote the vector whose jth coordinate is 1 and whose other coordinates are 0. Prove…

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Q: . Consider V1 1 and v3 = V2 = .0. 0 ) as vectors in R3. Let T: R3→R3 be a linear operator such that…

A: The solution is given as

Q: If |r(t)| = c (a constant), then r' (t) is orthogonal to r(t) for all t.

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Q: Let T:R2 R2 be a linear transfomation define as T T()=(;) and T(,) = G) Hind TC)? Find?

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Q: Show that the operators T and T₂ from 2 R² into R² de fined by T₁: (F₁) √₂ ) → (9₂9₁) and T2=…

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Q: Let ūi= 3. 5 %3D and u = Uz %3D Is V= 田。 e Span

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Q: Suppose uy, v2, Vm is linearly independent in V and w E V. Prove that dim span (v1 + w, vz + ,... "m…

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Q: Solve the following relations for x and y, and compute the Jacobian J(u,v). u= 2x + 2y, v = 3x -y…

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Q: Consider the linear operator defined by Show that T is compact. E1 E2 E3 T:l² l², x = (1, 2, 3, ...)…

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Q: Let LR→ R be a linear operator such that L X, )=| x, then Im(L) spanned by: Select one: a. (0,e,,e2)…

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Q: et X be a normed linear space and T: X→ X and S: X→ X be two perators, then S T :X→X is a bounded…

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Q: Let L:R R be a linear operator such that L X2 X3- X2 then Ker(L) spanned by: Select one: (e, e )

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Q: Let L:R → R3 be a linear operator such that L( X2 X1 then Im(L) spanned by: X3 X1 Select one: a. O…

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Q: Show that the operators Ix=x 0x = 0 Tx (t)=n't & are Linear. 6

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Q: Determine whether the following are linear operatorson Rn×n. L (A) = A + I

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Q: Consider the D-operator P(D) = mD² + 2kD +¢, where m and c are positive. If -c/m < k< 0, then

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Q: Let X ={li2,3,4,53 Determime the tnith valve Of each of the pioparitons. %3D a) (bx EX)x + x? <3Q t.

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Q: Let f(x, y) = 2x² + 5xy + 2y² + 5x – 5y + 2. Find the linearization L(x, y) of f(r, y) at the point…

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Q: Let LR R be a linear operator such that L X, )-| X1 then Ker(L) spanned by: Select one: a. (e2,e3)…

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Q: find the linearization at x = a. y = (1 + x)^−1/2, a = 0

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Q: Verify, if u = x + y+ z, v = x° + y° + z³ – 3ryz, w = x²+y² + z? – xy – yz – za are functionally…

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Q: Let T: X→→→→→ Y be a linear operator and dim X = dim Y=n<∞. Show that R(T) = Y if and only if T¹…

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Q: Let f, g, ϵ F (R , R) be the functions defined by f (t) = ert and g(t) =est, where    r ≠ s. Prove…

A: We can prove this question by the contradiction method, So, Assume that the opposite. Let {f, g}…

Q: (b) Show that the Linear guen Operators. are not with T(f)= Lf6), f()+1].

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Q: Let F: R: R? be defined as F(x, y) = %3D What is Jacobian matrix of F(r, y) at the point (1, 1)? 2 2…

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Q: The linear operator T: R2 --> R2  defined by T (x; y) = (4x + 5y; 2x + y) is it diagonalizable?

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Q: If F.T ds = 0, then F and T are orthogonal. %3D C O True False

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Q: Say the linear operator given T:R³ → R³, defined by T(x, y, z) = (x + y, x – y + 2z, 2x + y – 2). I…

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Q: (d) Let A E M3x3(R) and , ij, Z ɛ R* and they are linearly independent. Suppose that we have A = ,…

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Q: find the linearization at x = a. y = (1 + x2)^−1/2, a = 0

A: We find the derivative of the given function using power rule and chain rule

Q: Show that the operators T,··, T4 from R² into R² defined by ... (§i, 52)→ (Š1,0) (§1, É2)→ (0, E2)…

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Q: Consider the linear operator T: R? → R² defined by the following equations. w1 = 3x1 + 11x, W2 = X1…

A: we have to find the inverse of T          (i.eT-1(w1,w2))where T:R2→R2 is given by…

Q: s the linear operator T : R3 --> R3 defined by T (x, y, z) = (2x - 3y + 7z, 5y + z, -z)…

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Q: a bonneled linear operator. Then prve. that. Kes's is clased. b) let 8:X>y be a

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Q: Let L:R R be a linear operator such that L X2 X 1 then Im(L) spanned by: Select one: a. Ob. (0,e.e)…

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Q: Let L: R→ R be a linear operator such that L X2 )= X3-X2, then Ker(L) spanned by: X3 X3-X1. Select…

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Q: Let L: R→ R be a linear operator such that L X2 )= X3-X2, then L Select one: O a. b. O d.

A: Solution of question as follows

Q: 8. Show that an isometric linear operator T: H→→ H satisfies T*T= I, where I is the identity…

A: Let T : H → H  be an isometric linear operator. We need to show that , T satisfies T* T = I We know…

Q: Let T : R → R be the linear operator given by T(a, b, c) = (a, 2b, 3c) . Then det T = O5. O 2. 6. O…

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Q: 6. If u = given by x = +2, where x+y+z= 1, prove that the stationary value of u is y b a+b+c' a…

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Q: Let T : R² → R³ be the linear map given by [2r1 + 7x2] 2i + 22 Write does the matrix A associated…

A: The given linear map is T:ℝ2→ℝ3Tx1,x2=2x1+7x2, x1-x2, x1+x2 To find the matrix A associated with T…

Q: Let L: R → R be a linear operator such that L X2 )=| X1 then Ker(L) spanned by: Select one: а. b.…

A: Given a linear operator L from R3 to R3. We need to find the set of vectors that spans Ker(L)

Q: a) V = R", (x, y) = (Ax) · (Ay) where A e Mn,n is invertible.

A: Given that, V∈Rn,  A∈Mn,n is invertible matrix, i.e  A≠0 we have to show that x,y=Ax.Ay is inner…

Q: Let T:R→R² be a linear operator defined by T(x,y)= (w,w,) where w, =x –-y and W, =-2r- y. %3D %3D…

A: T0Tx = T(t(x))

Q: Compute the Jacobian of the mapping. G(r, t) = (2re", 3 + 8e")

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Q: Conversely, if T :V V is a linear operator such that T2 = T, show that V = kerT O imT. [Hint: 3.b, v…

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Q: Let L:R→ R be a linear operator such that L X2 )=x1, then Im(L) spanned by: -> X3 Select one: a. 一 O…

A: The given linear operator is: L:ℝ3→ℝ3 such that Lx1x2x3=x1x1x1