Question

a)Suppose U is a nonempty subset of the vector space V over field F. Prove that...

a)Suppose U is a nonempty subset of the vector space V over field F. Prove that U is a subspace if and only if cv + w ∈ U for any c ∈ F and any v, w ∈ U

b)Give an example to show that the union of two subspaces of V is not necessarily a subspace.

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
For a nonempty subset S of a vector space V , define span(S) as the set...
For a nonempty subset S of a vector space V , define span(S) as the set of all linear combinations of vectors in S. (a) Prove that span(S) is a subspace of V . (b) Prove that span(S) is the intersection of all subspaces that contain S, and con- clude that span(S) is the smallest subspace containing S. Hint: let W be the intersection of all subspaces containing S and show W = span(S). (c) What is the smallest subspace...
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}...
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 U1, U2 be subspaces of a vector space V. Prove that the union of U1...
Let U1, U2 be subspaces of a vector space V. Prove that the union of U1 and U2 is a subspace if and only if either U1 is a subset of U2 or U2 is a subset of U1.
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
Let U and W be subspaces of a nite dimensional vector space V such that U...
Let U and W be subspaces of a nite dimensional vector space V such that U ∩ W = {~0}. Dene their sum U + W := {u + w | u ∈ U, w ∈ W}. (1) Prove that U + W is a subspace of V . (2) Let U = {u1, . . . , ur} and W = {w1, . . . , ws} be bases of U and W respectively. Prove that U ∪ W...
Suppose U and W are subspaces of V. Prove that U+W is a subspace of V.
Suppose U and W are subspaces of V. Prove that U+W is a subspace of V.
Let U be a vector space and V a subspace of U. (a) Assume dim(U) <...
Let U be a vector space and V a subspace of U. (a) Assume dim(U) < ∞. Show that if dim(V ) = dim(U) then V = U. (b) Assume dim(U) = ∞ and dim(V ) = ∞. Give an example to show that it may happen that V 6= U.
Let U and W be subspaces of a finite dimensional vector space V such that V=U⊕W....
Let U and W be subspaces of a finite dimensional vector space V such that V=U⊕W. For any x∈V write x=u+w where u∈U and w∈W. Let R:U→U and S:W→W be linear transformations and define T:V→V by Tx=Ru+Sw . Show that detT=detRdetS .
Let S, U, and W be subspaces of a vector space V, where U ⊆ W....
Let S, U, and W be subspaces of a vector space V, where U ⊆ W. Show that U + (W ∩ S) = W ∩ (U + S)
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT