7. Let X be a topological space, let A C X be a subspace, and let i: A → X the inclusion map. Fix a basepoint ao E A, and consider the induced homomorphism on fundamental groups, i*: ₁(A, αo) → π₁ (X, αo). (a) Suppose A is a retract of X. Show that i is injective. (b) Give an example of an inclusion i: A → X where i is not injective.

Q: (g) A subset ACX is said to be a weak deformation retract of X if the inclusion map i: A X is a…

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Q: Let fg,:I→R2 begivenby f(x) = (0, x), g(x) = (1, x). a. Specify a homotopy h:I × I→R2 from f to g.…

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Q: 9. Let X = {(x, y, z): √x² + y² + z² ≤ 1} be a subspace of R³ with the standard topology and 0 = (0,…

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Q: 62(6):Let T: R> R be a a lineas, transfe defined by T(x= (x-y,x,x+24). let "B, B be l ordered basis…

A: Solution: The linear transformation is Tx,y=x-y,x,x+2y

Q: Determine: 1 [1 -1 2]}-1 [2 3]+|0 1 -1

A: Step:-1 Given that  1-124-1023+1001-1-1Take U=1-124-1023+1001-1-1Since, we…

Q: 15. If T is a self-adjoint linear operator on a finite-dimensional inner product space V, then detT…

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Q: estion 1 Consider the groups Z² = Z × Z and Z³ = Z× Z× Z and the map ƒ : Z³ → Z² ined by the rule…

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Q: Example (3): suppose X = F", with the inner product defined as (x, y) = E=1*iYt, Vx, y'E X, prove…

A: The objective is to prove that X is Hilbert space.

Q: Suppose A is a normed algebra and define the map MB : A → A by MB(A) = BA. Then Mg is continuous on…

A: A is normed algebra Mb: A→AMb(A)=BA

Q: Prove that if X is a Banach space, X ∕= {0}, then there is a non-zero biounded linear functional on…

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Q: [1] Given the homomorphism p: P4 - R3 defined by [b + c] p(a + bx + cx2 + dx3 + ex4) = d La Find the…

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Q: 4. Let X = {(t,co): te R} U {[t, oo): te R}. Consider the binary relation C on X, that is, a Cy if…

A: Given: X = {(t,∞) : t ∈ R} U [ t , ∞) : t ∈R }. Consider the binary relation ⊆ that is, x ⊂ y if and…

Q: For all v € V, let : V' → F be defined by (p) = y(v) for y € V'. (a) Show that is linear, i.e. € V",…

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Q: 2. Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel?…

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Q: Let φ : G → H be an onto homomorphism.(a) Assume that G is abelian. Does this imply that H is…

A: Let's analyze each part of the question: (a)  Assume φ: G → H is an onto homomorphism and G is…

Q: Let T be a bounded operator on a Banach space X and let Q e an isomorphism on X. Show that eTQ=…

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Q: Let T : V → U be a homomorphism, then Ker T = { 0 } iff T is one-one

A: A linear transformation is said to be injective or one to one if provided that for all v1 and v2 in…

Q: Suppose X, Y, Z, X', Y', and Z' are A-modules. Consider the com- mutative diagram with exact rows: X…

A: Suppose X,Y,Z,X',Y' and Z' are A-modules.

Q: Let k: Y → Z to be a map; • Prove that if fo, fi: X → Y are homotopic maps, then ko fo, ko fi: X Z…

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Q: 3. Suppose V is finite-dimensional and S, T are linear operators on V. Prove that ST is invertible…

A: Proof are in given below Explanation:Step 1:

Q: {( º ) | a,b,c € z}. (a) Show a 0 b c that L is a subring of M₂(Z). (b) Show that the map f: L→ Z…

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Q: Let V = V(r - y) C A. Consider the morphism o: A' → V given by o(t) = (t,t). (a) Show that o is a…

A: Given: V=Vx3-y2⊂A2 ϕ:A1→V by ϕt=t2,t3 To do: Show that ϕ is a bijection but not an isomorphism.…

Q: 1. Let X be a topological space and be an equivalence relation on the elements of X. Define : XX/~…

A: Given: πx=x To find: Solution

Q: Compute ø(25) for the homomorphism ø:Z→S, such that p(1)= (1,4,2,6)(2,5,7) (1,6)(4,7) O None of them…

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Q: Let M, and M2 be two R-modules and N, and N2 be two submodules of M, and M2 respectively. Also, let…

A: Here we need just one fact that if f:X→Y is a function defined from X to Y and if y∈Y, then f-1(y)∈X…

Q: Let M and M2 be two R-modules and N, and N2 be two submodules of M, and M2 respectively. Also, let…

A: We will use fundamental theorem of homorphism here . If f : G-->G' is onto homorphism then G/kerf…

Q: Let T be an operator on an innner product space V, and suppose that ||T(x)|| = ||x|| for all x in V.…

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Q: Let C(X,Y) have the compact-open topology. Show that if Y is Hausdorff, then C(X, Y) is Hausdorff

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Q: Exercise 7. Let X be a topological space. Suppose B(x) is a basis of neighborhoods of x = X. a) For…

A: You have asked more than one question which itself are single questions and have broad answer.…

Q: linear transformation from a finite dimensional inner product space V to itself is shew ymmetric if…

A:

Q: Suppose char F = 0. Then F contains a subring S such that S2 (i.e. there is an somorphism f: S→ Z).…

A: Given That : Let char F=0 then F contains S and S be a subring such that S≅Z that is f is an…

Q: Suppose we have ki(x, z) is a valid kernel. Then the following is also valid kernel: k(x; z) =…

A: The kernel trick provides a solution to this problem. The “trick” is that kernel methods represent…

Q: Let {A,i e I} be a set of submodules of a module M and let B M. ( a) Prove Σ (A,ο Β)- Σ.)B.

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Q: 6. Classify the given mapping y: A→B by checking 5 properties. Each property must be explained. y :…

A: Sol:- The given mapping y: A → B, where A = (−∞, +∞) and B = [−5, +∞), can be classified as follows:…

Q: For a continuous transformation ƒ : X → X, prove that f is topological- transitive if and only if…

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Q: 1. Let O1 and O2 be topologies on X. (1) Show that the identity map idX : (X, O1) → (X, O2) is…

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Q: ► Exercise 11.2. Prove Proposition (c) A finite product of covering maps is a covering map.

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Q: Define T: C² C² by T((w, z)) = (iw, z-w). Show that T is an isomorphism, s, show that T is linear,…

A: We need to show that T is linear,oneone,onto

Q: Let φ: ℤ→ℤ defined by φ(x) = x + 2. Show φ is a homomorphism.

A: We must demonstrate two characteristics, preservation of addition and preservation of identity, in…

Q: Let f be non-negative measurable function on IR. show that   limn→∞ ∫[-n,n] f = ∫IR f

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Q: 1. Let X be a normed space. Show that the addition (x, y) = XxXx + y E X and the scalar…

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Q: 8. Let A be an n x n matrix over a field K. Show that the mapping f defined by f(X,Y)= XT AY is a…

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Q: Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the following…

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Q: 5. Show that the function that sends an ordered pair ((x1, x2, x3), (Y1, Y2, Y3)) of elements in R³…

A: I have write answers below Explanation:

Q: Exercise 3. Prove that a relation R on A is antisymmetric if and only if RNR-' C {(a, a) | a E A

A: We know that , A relation R on set A is antisymmetric  if either x , y∉ R  or (y , x) ∉ R whenever…