Let G be a direct product of H and K then Z(G) is a direct product of Z(H) and Z(K).

Q: 1. Describe the transformation(s) from f(x) = x2 to g (x) = (x – 2)2 +4 %3D

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Q: Simplify generator of I = (x – 2y, Y – 3z, z – 4x) as ideal of R[r, y, z], respectively of J = (x –…

A: Prove that all the variables belong to I. Then conclude the value of I.

Q: True or False: If f and g are both invertible (i.e. have their inverses), then f + g is also…

A: Given the statement that if f and g are both invertible functions then their sum (f + g) is also…

Q: Use Gauss-Jordan method to calculate the inverse of A A-l = |1 0 -1 A =| 0 1 1 1 1 1

A: By Gauss Jordan Elimination method the inverse of matrix is given as follows :

Q: Show that if f is a rational function, and if g is a fractional linear transformation, then f and…

A: consider the equation

Q: Show that there does not exist a linear transformation from R’ to R² whose kernel equals {(x1, x2,…

A: We can solve this by  the help of rank nullity theorem

Q: Prove the product rule by use of ΔA=w⋅Δh+h⋅Δw+Δw⋅Δh

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Q: Let -6 2 A = 2 -3 Then the spectral decomposition is given by:

A: In the given question we have to find the matrix which gives the spectral decomposition of the…

Q: What can you say about the dimensions of A and B_if both products AB and BA are defined, although…

A: If both products AB and BA are defined then dimensions of A and B will be equal and yes, it is not…

Q: х1 — 2х2 -x1 +3x2 3x1 – 2x2 - x1 Let T : R2 → R be a linear transformation such that T( x2 X1 Find i…

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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: Let Pn be the set of polynomials of degree at most n and R be the set of real numbers. L: Pn → R…

A: A transformation 'T' is said to be linear if  T(a + b)= T(a) + T(b)

Q: Let y be a finite dimensional Linear product

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Q: Prove that Z3 x Zz is isomorphic to Zg.

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Q: Use Fourier-Motzkin elimination to give a linear combination of the following in- equalities that…

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Q: Use the inverse form of the first translation theorem to determine the power of e. 2! %3D (S-5)° -5t…

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

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Q: Describe the group of the polynomial (x4 – 5x2 + 6) ∊ ℚ[x] over ℚ.

A: Please see the next step for solution

Q: Consider the transformation L: R? → R² given by =(;)-[: :|[:) 0 0 0 1 L What is the kernel of L?

A: We will find Kernel which is null space

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: 223+-5 -j Suppose z1 = 2+ j, 2 = 3- j2 and z3 = -+j Evaluate %3D 2-a+3-j

A: Substitute the given values of z1,  z2, z3 in the given expression and simplify.…

Q: Find Fourier transformation of f(x) = (1-x) x<1

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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: Transform fx) into g(x) where the transformation is g(x) = 2 · Ax - 2)

A: It is given that the line segment joining from (0,0) to (-1,2). This line segment is a function…

Q: 3. Determine whether the linear transformation x → Ax = (x + x2 + x3,x1+ x3, x2 – 13), x = (*1, x2,…

A: Use matrix determinant to find invertibilty of given linear transformation and also find matrix…

Q: 2. Determine all k e R for which A is diagonalizable. [1 k 1 (a) A [1 0 k (b) A= |0 1 0 0 0 k

A: Solve the following

Q: Find the Wronskian for {x2, ex^2, x2ex

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Q: Show that the transformation T(x1, x2) = (2x1 − 2x2, 9x1 − 8x2) is invertible, and find a formula…

A: Consider the transformation as,

Q: Let G be a direct product of H and k product of 2 (H) 'and z (k) then Z(G) is a direct

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Q: True or False: (p, q) = pq defines an inner product on P(R).

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Q: For what values of a does the homomorphism system (b – 1)x + 2y = 0 2x + (b – 1)y = 0 Have a…

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Q: Let I (f (x)) be the principal ideal generated by f(x) in Z2[x]. Calculate the multi- plicative…

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Q: When given the Linear Transformation formula as follows:  P : R³ to R³, where P[x1,x2,x3] =…

A: We have to find the matrix PQ and QP.

Q: Prove that L(x) = (2x1 – x2, X2 + x3)" is a linear transformation.

A: To prove that L(x) = L( x1 , x2 , x3 ) = ( 2x1 - x2 , x2 + x3 ) is a linear transformation.

Q: Determine whether the linear transformation T: R3→R3 defined by T(x1, x2, x3) = (x1 + x2, x2 + x3,…

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Q: のュキx +3x-2ズュズs [An:3,-+2+チ7

A:

Q: Prove that is G1 is isomorphic to G2 and H1 is isomorphic to H2, then G1 x H1 is isomorphic to G2 x…

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Q: Find Fourier transformation of f(x): [(1-x) |x|1

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Q: Show that if f and g are multiplicative functions, then the Dirichlet product f g, defined by (f…

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Q: (c) Prove that G/N is Abelian iff aba-¹b-¹ N a,b G. 2

A: We have to prove the given theorem.

Q: The elementary symmetric polynomials in three variables are given by: S1(х, у, z) %3D х + у+ z,…

A: Firstly, we note that  is the expansion of  in terms of the symmetric polynomials This is a well…

Q: For z=1+i find the product of z and its complex conjugate zZ

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Q: Let Qo: R2 R2 be reflection in the x-axis, Q₁: R² R² be reflection in the line y = x, let Q-1: R² R²…

A: Let 1) Q0:ℝ2→ℝ2, be the reflection in the x-axis  2) Q1:ℝ2→ℝ2 be the reflection in the line y=x. 3)…

Q: Let a ∈ R[[x]] with a0 not equal to 0. Prove that a has a multiplicative inverse in R[[x]]. You may…

A: We have given that , a∈ℝx a = a0 + a1x + a2x2 + ......  where each ai ∈ ℝ. Since , 1 ∈ ℝx  is an…

Q: Let z (х, у). Show that z- 2' 2 X + y -

A: We have to find the multiplicative inverse of z.

Q: 7. Find the inverse of the following linear transformation: [3x + 2y] 5x +3yl

A: 7 Given linear transformation is Txy=3x+2y5x+3y

Q: Find the image of | z – 3i | = 3 under the mapping W =-.

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Q: 2} and {bn} are divergent then {an + bn} is divergent.

A: Given that: an and bn are divergent.

Q: Show that the transformation :R2 →  R3 defined by  T x X1 X2 = X1+X2 3X2 X1-X2 is…

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.