Question

Let (V, |· |v ) and (W, |· |w ) be normed vector spaces. Let T...

Let (V, |· |v ) and (W, |· |w ) be normed vector spaces. Let T : V → W be linear map. The kernel of T, denoted ker(T), is defined to be the set ker(T) = {v ∈ V : T(v) = 0}. Then ker(T) is a linear subspace of V .

Let W be a closed subspace of V with W not equal to V . Prove that W is nowhere dense in V .

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
1. Let T be a linear transformation from vector spaces V to W. a. Suppose that...
1. Let T be a linear transformation from vector spaces V to W. a. Suppose that U is a subspace of V, and let T(U) be the set of all vectors w in W such that T(v) = w for some v in V. Show that T(U) is a subspace of W. b. Suppose that dimension of U is n. Show that the dimension of T(U) is less than or equal to n.
Let V and W be finite-dimensional vector spaces over F, and let φ : V →...
Let V and W be finite-dimensional vector spaces over F, and let φ : V → W be a linear transformation. Let dim(ker(φ)) = k, dim(V ) = n, and 0 < k < n. A basis of ker(φ), {v1, . . . , vk}, can be extended to a basis of V , {v1, . . . , vk, vk+1, . . . , vn}, for some vectors vk+1, . . . , vn ∈ V . Prove that...
Construct a linear transformation T : V → W, where V and W are vector spaces...
Construct a linear transformation T : V → W, where V and W are vector spaces over F such that the dimension of the kernel space of T is 666. Is such a transformation unique? Give reasons for your answer.
Let L : V → W be a linear transformation between two vector spaces. Show that...
Let L : V → W be a linear transformation between two vector spaces. Show that dim(ker(L)) + dim(Im(L)) = dim(V)
1. Let V and W be finite-dimensional vector spaces over field F with dim(V) = n...
1. Let V and W be finite-dimensional vector spaces over field F with dim(V) = n and dim(W) = m, and let φ : V → W be a linear transformation. A) If m = n and ker(φ) = (0), what is im(φ)? B) If ker(φ) = V, what is im(φ)? C) If φ is surjective, what is im(φ)? D) If φ is surjective, what is dim(ker(φ))? E) If m = n and φ is surjective, what is ker(φ)? F)...
Suppose V and W are two vector spaces. We can make the set V × W...
Suppose V and W are two vector spaces. We can make the set V × W = {(α, β)|α ∈ V,β ∈ W} into a vector space as follows: (α1,β1)+(α2,β2)=(α1 + α2,β1 + β2) c(α1,β1)=(cα1, cβ1) You can assume the axioms of a vector space hold for V × W (A) If V and W are finite dimensional, what is the dimension of V × W? Prove your answer. Now suppose W1 and W2 are two subspaces of V ....
5. Prove or disprove the following statements. (a) Let L : V → W be a...
5. Prove or disprove the following statements. (a) Let L : V → W be a linear mapping. If {L(~v1), . . . , L( ~vn)} is a basis for W, then {~v1, . . . , ~vn} is a basis for V. (b) If V and W are both n-dimensional vector spaces and L : V → W is a linear mapping, then nullity(L) = 0. (c) If V is an n-dimensional vector space and L : V →...
Let V and W be vector spaces and let T:V→W be a linear transformation. We say...
Let V and W be vector spaces and let T:V→W be a linear transformation. We say a linear transformation S:W→V is a left inverse of T if ST=Iv, where ?v denotes the identity transformation on V. We say a linear transformation S:W→V is a right inverse of ? if ??=?w, where ?w denotes the identity transformation on W. Finally, we say a linear transformation S:W→V is an inverse of ? if it is both a left and right inverse of...
Suppose V is a vector space over F, dim V = n, let T be a...
Suppose V is a vector space over F, dim V = n, let T be a linear transformation on V. 1. If T has an irreducible characterisctic polynomial over F, prove that {0} and V are the only T-invariant subspaces of V. 2. If the characteristic polynomial of T = g(t) h(t) for some polynomials g(t) and h(t) of degree < n , prove that V has a T-invariant subspace W such that 0 < dim W < n
(3) Let V be a finite dimensional vector space, and let T: V® V be a...
(3) Let V be a finite dimensional vector space, and let T: V® V be a linear transformation such that rk(T) = rk(T2). a) Show that ker(T) = ker(T2). b) Show that 0 is the only vector that lies in both the null space of T, and the range space of T