Question

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.

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
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 .
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 T be a 1-1 linear transformation from a vector space V to a vector space...
Let T be a 1-1 linear transformation from a vector space V to a vector space W. If the vectors u, v and w are linearly independent in V, prove that T(u), T(v), T(w) are linearly independent in W
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...
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)
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...
Let T:V→W be a linear transformation and U be a subspace of V. Let T(U)T(U) denote...
Let T:V→W be a linear transformation and U be a subspace of V. Let T(U)T(U) denote the image of U under T (i.e., T(U)={T(u⃗ ):u⃗ ∈U}). Prove that T(U) is a subspace of W
Let T: U--> V be a linear transformation. Prove that the range of T is a...
Let T: U--> V be a linear transformation. Prove that the range of T is a subspace of W
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
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 ....