7. Let X be a topological space, let A CX be a subspace, and let i: AX theinclusion map. Fix a basepoint ao E A, and consider the induced homomorphismon fundamental groups, i: 7₁(A, ao) → л₁(X, ao).(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.

Basics Inclusion Map: Given a subset B of the set A, the injection f:B→A defined by f(b)=b for all…

Answered: 7. Let X be a topological space, let A…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 6CM: Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a...

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Similar questions

11. Show that defined by is not a homomorphism.
Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.
Find all homomorphic images of the quaternion group.
Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a the kernel of T and b the range of T. c Determine the rank and nullity of T.
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Let T be a linear transformation from R2 into R2 such that T(1,2)=(1,0) and T(1,1)=(0,1). Find T(2,0) and T(0,3).
3.) Let V be a finite-dimensional vector space. Suppose T : V V is a linear transformation. Show that T is injective (one-to-one) if and only if T is surjective (onto).
Let V be a finite-dimensional inner product space over C, and let S, T be linear maps V + V. (i) Define the adjoint map T* : V + V. (ii) Show that (ST)* = T*S*. (iii) Prove that if T is invertible, then T* is also invertible and (T*)-1 = (T-1)*.
5. 3) Let : R₂[x] → R³ be a linear transformation such that ( (1) = = l(x): = 0 and l(x²) = = 1 (a) Find a formula for l(a + bx + cx²). (b) Show that is a vector space isomorphism and find a formula for l-¹.
3. Let V and W be n and m dimensional F-vector spaces and let B and A be bases for V and W respectively. Let YB V → Fn and A: W→ Fm denote the coordinate isomorphisms. Let S V W be a linear transformation. (a) Prove that 7 maps Ker S onto null[S] and 7 maps imS onto col[S]. 2 (b) Conclude that null[S] ≈ (ker S) and col[S] ≈ imS are isomorphisms. (c) Write a statement that connects rank and nullity of [S] to the kernel and image of S.

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