Question

Consider vector spaces with scalars in the field F(could be R or C). Recall that L(V,...

Consider vector spaces with scalars in the field F(could be R or C). Recall that L(V, W) is the vectors space consisting of all linear transformations from V to W.

a. Prove that L(F, W) is isomorphic to W.

b. Assume that V is a finite dimensional vectors space. Prove that L(V, F) is isomorphic to V.

c. If V is infinite dimensional, what happens to L(V, F)?

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
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 L : V → W be a linear transformation between two finite dimensional vector...
) Let L : V → W be a linear transformation between two finite dimensional vector spaces. Assume that dim(V) = dim(W). Prove that the following statements are equivalent. a) L is one-to-one. b) L is onto. please help asap. my final is tomorrow morning. Thanks!!!!
4. Prove the Following: a. Prove that if V is a vector space with subspace W...
4. Prove the Following: a. Prove that if V is a vector space with subspace W ⊂ V, and if U ⊂ W is a subspace of the vector space W, then U is also a subspace of V b. Given span of a finite collection of vectors {v1, . . . , vn} ⊂ V as follows: Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in the scalar field}...
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 ....
1. Assume that V is a vector space and L is a linear function V →...
1. Assume that V is a vector space and L is a linear function V → V. a. Suppose there are two vectors v and w in V such that v, w, and v+w are all eigenvectors of L. Show that v and w share the same eigenvalue. b. Suppose that every vector in V is an eigenvector of L. Prove that there is a scalar α such that L = αI.
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 U and V be subspaces of the vector space W . Recall that U ∩...
Let U and V be subspaces of the vector space W . Recall that U ∩ V is the set of all vectors ⃗v in W that are in both of U or V , and that U ∪ V is the set of all vectors ⃗v in W that are in at least one of U or V i: Prove: U ∩V is a subspace of W. ii: Consider the statement: “U ∪ V is a subspace of W...
Let V be a finite-dimensional vector space and let T be a linear map in L(V,...
Let V be a finite-dimensional vector space and let T be a linear map in L(V, V ). Suppose that dim(range(T 2 )) = dim(range(T)). Prove that the range and null space of T have only the zero vector in common
3. Let V and W be finite-dimensional vector spaces over field F with dim(V) = n...
3. 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. Fill in the six blanks to give bounds on the sizes of the dimension of ker(φ) and the dimension of im(φ). 3. 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...
Let V be a finite-dimensional vector space over C and T in L(V) be an invertible...
Let V be a finite-dimensional vector space over C and T in L(V) be an invertible operator in V. Suppose also that T=SR is the polar decomposition of T where S is the correspondIng isometry and R=(T*T)^1/2 is the unique positive square root of T*T. Prove that R is an invertible operator that committees with T, that is TR-RT.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT