Let f : A → B and g : B → C be bijections. Prove that gof is invertible and that (g o f)-1 = f-1 o g¬1.

Q: Prove Theorem (A + B )T = AT + B T

A: Prove Theorem (A + B )T = AT + B T

Q: . Show that the operators T₁,, T4 from R2 into R2 defined by (1, 2) (1, 0) (§1, §2) (0, §₂) (§1, §2)…

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Q: Let T: P4(R) → P4(R) be defined by T(f(x)) = xf"(x). Find a Jordan canonical form of T.

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Q: Let T : M2x2(R) → P2(R) be defined by T с d = (a + b) + (2d)x+ bx². Let B = [(6 ") (C ) G ) C ) 0 0…

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Q: Let xe c and ye show that <xiy= y AX is qu inmer product if A is hermition positive del iniite

A: (Solving the only one question as per our guidelines) Question: Let x∈ℂm and y∈ℂm Show that x,y=y*Ax…

Q: Give V = C([0, 1]) the inner product defined as follows: (f, g) :| f(t)g(t) dt for all f, g e V.…

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Q: Let T be self-adjoint on a finite-dimensional inner product space V. Prove that Vr E V, ||T(x)± i||?…

A: The complete solutions are given below

Q: Assume A is nonsingular. Prove that det(A-1) 1 det(A) Let c be any scalar. Prove that det(cA) = c"…

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Q: suppose Vn is the vector space of polynomials of degree <n-1 and with real cofficients. Determine…

A: As per the question we are given a vector space Vn of polynomials with degree ≤ n-1 and with real…

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Q: If z = f(u) + g(u), where u = - and v = x y. Prove that y %3D x: Zx - y· Zy = y2 · Zyy – x2 . Zxx…

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

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

A: Here, the given function is: f(x,y,z)=2x2-y2+3z2

Q: Let fG) = x2+I E Z, x] and let R= Z, [x]/I, where I= (f (x)) @ Show that R is a field with 9…

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Q: Let T: R" → R" be defined as T(x₁,x₂,...,xn) = (x₂ −X₁, X3 X₂, ..., x₁ - xn) of T Discuss the…

A: The matrix is invertible if the determinant of the matrix is non-zero. The matrix is diagonalizable…

Q: Prove that if A is invertible, then AT is invertible. What is (AT)-1?

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Q: Prove that o curl f dS = | 6f · dr - v ¢ x f · d$.

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Q: Prove the following relation in the vector triple product : Ax(BxC) + Bx(CxA)+Cx(A x B) = 0

A: The answer is given in handwritten form in the step-2 box. Please check it.

Q: Prove that if u is orthogonal to both v and w, then u is orthogonal to sv+tw for all scalars s and…

A: Given that u is orthogonal to both v and w.

Q: Determine a scalar r such that Ax = rx, where

A: Matrix: A=12-11014-45x=-12141AX=12-11014-45.-12141

Q: Consider f, g E T° (R") and 0 > 0. Show the following results on the subdifferential. (a) a(0f)(x) =…

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Q: EXERCISE 6.1. Let f: XY and g: Y Z. Prove that gof:X-Z is a bijection.

A: Given : f : X→Y and g : Y→Z To prove : g∘f : X→Z is a bijection We know that, a bijection mapping is…

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: Let H be a Hilbert space and A be a positive operator on H. Prove that I + A is invertible in BL…

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Q: Let f : A → B and g : B → C be bijections. (a) Prove that (go f)-1 exists. (b) Prove that (g o f)-1…

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Q: Let u,v,w be linearly independent over a field F.  Show that{u+v,u−2v,u−v−2w}is also linearly…

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Q: im(T1 + 14T) im(14T† + T2).

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Q: The mapping w = z² + 1 is not conf ormal at O z = 0 O z = i O z = -i

A: We know that analytical function is not conformal at critical points. Let's find the critical points…

Q: Let E/F be Galois and f e F[r] irreducible. Prove that any two irre- ducible factors of f € E[r]…

A: Given E /F

Q: Let X be a Hilbert space and T:X → X and T2: X → X are self-adjoint operators such that either…

A: Given we have given that X is a Hilbert space and T1 : X→X &amp; T2 :X→X  are self -adjoint…

Q: prove that A is invertible and A = B = C. %3D

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Q: Determine if the polynomials p(t) = 2 - t, q(t) = t? + 2t are orthogonal under the inner product…

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Q: at x and y are orthogonal

A: Given: x and y in ℝn, x+y=x-y To show:  x and y are orthogonal in ℝn if and only if x + y = x − y

Q: Let E/F be Galois and f e F[r] irreducible. Prove that any two irre- ducible factors of f € E[x]…

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Q: Let r; = ro + hj for j = 0,1,2. Use the quadratic polynomial interpolating f at those nodes %3D %3D…

A: The solution is given below

Q: Show that if k is a scalar and A is n × n, then det(kA) = kn det(A).

A: Show that if k is a scalar and A is n × n, then det(kA) = kn det(A). For any scalar k, we need to…

Q: Let X be co or lº, 1 ≤ p ≤ ∞o. Consider the operator Prove that U is not compact but U² is compact.…

A: Please find the answer in next step

Q: Now Considex a fn abarce of set of finctrons f() x€ (-00gt00) such that of d del varieble Oes ox→ I…

A: As per the instruction, we will be answering only the second part. For (ii), We consider a function…

Q: Use the inner product 1 (f, 8) = f(x)g(x) dx in the vector space CO[0, 1] of continuous functions on…

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Q: Let xe C and ye show that <xiy= y* AX is que inmer product if A is hermitian positive def iwite

A: Let x∈ℂm and y∈ℂm To prove that x,y=y*Ax is an inner product if A is Hermitian positive definite.…

Q: Show that the function defines an inner product on R2, where u = (u1, u2) and v = (v1, v2). ⟨u, v⟩ =…

A: Given function u,v=u1v1+9u2v2 where u=u1,u2 and v=v1,v2 The function is defined on R2. To show that…

Q: Let T be a self-adjoint operator on a finite-dimensional inner product space V. Prove that for all x…

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Q: Show that each linear operator T on R2 is nonsingular and find a formula for T- , where (a) T(x,y) =…

A: We have to show that linear operator T on R2 is nonsingular andfind the formula for…

Q: Let R be the ring of continuous functions from R to R. Prove that M = {f e R : f(0) = 0} is a…

A: To show that the given subset M of R is a maximal ideal, we first show that M is a ideal of R and…

Q: b) Let f(x) = 1 and g(x) = x be functions in C[0, 1]. Use the inner product defined in Example 5, to…

A: Note: Since there is no proper instruction is given and the first part is incomplete, we have solved…

Q: Exercise 16:  Prove the following: Theorem 30: Let a, b ∈ Z where a is not equal to 0. Then Rem(b,…

A: Solution:

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

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Q: Let V be a finite-dimensional inner product space, and let T be a linear operator on V. Prove that…

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Q: For a linear operator T on an inner product space V, prove that T∗T = T0 implies T = T0. Is the same…

A: Let T be a linear operator defined on an inner product space V.To prove

Q: Determine a scalar r such that Ax = rx, where

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