Let R→R be defined by g(x) 2y+1 2y+2 Is g bijective? Prove or give a counter example.

Q: Verify that the Galois imaginary is primitive in GF(3, x³ + 2x+1)

A: Given:- Verify that the Galois imaginary is primitive in GF(3,x3+2x+1)

Q: Show that the function a Y1 (x, y) = [21 2] в с Y2 is an inner product on R? if and only if b2 – ac…

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Q: Let z = ƒ(u, y), u = ax + by, and y = ax - by. Express zx andzy in terms of fu , fy , and the…

A: Given, Let z=f(u,Y) u=ax+byY=ax-by

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: Is {x² – 2.x, x²+x, x²} linearly independent dependent? Prove your answer. or

A: Answer

Q: Draw the bifurcation diagram for i = (x – 3)(c+ x)

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Q: Let A={x e R|-1sx s1} = B, Show that f:A →B given by f (x)=x|x| is a bijecti on.

A: we have to show that given function is bijective

Q: Show that A = {xf(x)+ 2g (x) : f (x), g (x) e Z [x]} is not a principal ideal of Z [x] and so Z [x]…

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Q: The conjugate z of z=x+iy is z=x-iy

A: 4) Conjugate of a complex number: The complex conjugate is obtained by changing the sign of the…

Q: If A²=0, prove that A cannot be invertible.

A: Given,If A²=0, prove that A cannot be invertible.

Q: Let, T:R' → R³;T(x,y,z)=(2x+ y, y- z,2y+4z) 2. Find the rank and nullity of T.

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Q: Prove that f : R – {3} → R – {1}, defined by f(x) = is a bijection. x-1 x-3

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Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C° (R). x In x are linearly independent in…

A: Concept:

Q: Show that an is decreasing. %3D 2n +1 1.

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Q: If z = f (u) + g(v), where u = = and v = x ·y. Prove that Q1: x· Zx – y· Zy = y² · Zyy – x² · Zxx…

A: We have z=fu+gv, and u=xy  and v=x.y We need to use the chain rule and prove x.zx-y.zy=y2.zyy-x2.zxx…

Q: Draw the bifurcation diagram for à = (x + 1)(x² – 2.x +r).

A: We are given : Now, the bifurcation diagram for this is :

Q: Verify that the operation {x, y} = x1y1 − x1y2 − x2y1 + 3x2y2 where x = (x1, x2) and y = (y1, y2) is…

A: Given : The operation on ℝ2 is defined as  x, y = x1y1 - x1y2 - x2y1 + 3x2y2 ,  where x = (x1 , x2)…

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

A: Problem based on function.

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

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Q: Find the linearization L(x) at x = a. 1 f(x) = - x - a = 1 %3D -

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Q: Prove that for a E N, a always invertible in Z,2+3a+1·

A: We know that an element a∈ℤn is called invertible if there is an element b∈ℤn such that a+b≡0  mod…

Q: Show why the product {[x1, x2], [yı, y2]} = xıy2 does not satisfy the conditions for an inner…

A: See the detailed solution below.

Q: (ii) Verify Cauchy-Goursat theorem for f(z) = 5 sin 2z if C is the square with vertices at 1土も-1土。

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Q: Q3: a) Let ƒ : Z → Z be such that f(x) = Vx² – x + 1. If invertible, and if it is, what is its

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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: A function f(x,y) is homogeneous of degree 7 in x and y if and only if, __________.

A: Given problem is solved below.

Q: Define g : (-1, 1) → R by g(x) 1- x2 " Show that |(-1, 1)| = |R| by showing that g is a bijection

A: Concept Used: For a function f:A→B If f is a bijection, then |A|=|B| Consider the function g:(-1,…

Q: Let A ∈ M nxn(F). Show that A is diagonalizable if and only if At isdiagonalizable

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Q: Let T be the linear thansformation from R5 to Rª defined by X1 1] X2 X1 1 3 2 X2 -8 1 -2 2 0 -1 4 X3…

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Q: 1. Show that the definition (x1, x2) · (Y1, Y2) = x1Y2 is not an inner product on R?.

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Q: prove V injective if x1 ‡xn → v(x1) = v(xn)

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Q: Calculate A and B so that t B %3D 1+1 (1-z²)

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Q: Please solve

A: Consider the given function.

Q: Consider the following bijections on R: f (x) = - g(x) = x – 7 Then (f o g)¯'(x) =

A: Given, fx=-x,gx=x-7 Find the value of f∘g-1x.

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

A: We have to prove this

Q: Let T be the linear thansformation from R³ to R4 defined by 1 -8 1 X1 1 3 2 X2 X2 -2 2 4 X3 X3 3 0…

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Q: Let A,X EM, (C) such that A is positive semidefinit and AX = XA Show that A¹/2X = XA¹/2.

A: Introduction: Definite matrix is significant in studying quadratic forms. Also, in Lyapunov…

Q: Draw the bifurcation diagram for i = (x – 3)(c + x)

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Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = x + 1/x, a = 1

A: Given fx=x+1x

Q: 3.2. let f(x) = 2, x+4 xez, * 1 - 1 1/2/20 -74x45 -*>$ Approximate the foldoy using Trapezoid mule…

A: In order to approximate the integral: ∫abydx, the trapezoidal rule uses the formula:…

Q: Q3: Using Fredholm determinates, find the resolvent kernel of the kernel : k(x, t) = sin(x) –…

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Q: Z-1 %3D ,then prove that x2 + y? = (x-1)2+y2 (x+1)2 +y2 (1) If Z %3D Z+1

A: Topic- complex number Note:- as per our guideline if more than one question uploaded than only first…

Q: Show that f : R → R is bijective, where: f(x) %3D필

A: A function is injective if f(x)=f(y)⇒x=y. A function is onto if for every y in Y there exists a x in…

Q: Let ƒ : [0, 1] → R be given by f(1) = xr and g : R → [0, 1] be given by g(x) = Prove that |[0, 1]| =…

A: Cantor-Schrӧder- Bernstein Theorem: if there exist injective functions f: A → B and g: B → A between…

Q: Prove that if A is invertible and AB=0, then B=0.

A: Given:

Q: 4. Show that the function f : X → Y defined by f(X) = X U {a} vX €x is a bijection.

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Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C (R). x In x are linearly independent in

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Q: Let z = 2 - 3xy?+ x². Zay %3D Zyy

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Q: If Ø : R3 → R3 be defined as Ø (x,y,z) = (2x, 4x –y,2x+ 3yz) then show that Ø is invertible. %3D

A: Given that ϕ:R3→R3 is defined as ϕx,y,z=2x,4x-y,2x+3yz. Now, ϕ is invertible if ϕ is one-one and…