Question

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 .” If the statement is true, prove it. If the statement is false, give an example demonstrating this.

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 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...
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)
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}...
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.
Let V be a vector space and let U1, U2 be two subspaces of V ....
Let V be a vector space and let U1, U2 be two subspaces of V . Show that U1 ∩ U2 is a subspace of V . By giving an example, show that U1 ∪ U2 is in general not a subspace of V .
Let u, vand w be linearly dependent vectors in a vector space V. Prove that for...
Let u, vand w be linearly dependent vectors in a vector space V. Prove that for any vector z in V whatsoever, the vectors u, v, w and z are linearly dependent.
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...
Let V be an n-dimensional vector space. Let W and W2 be unequal subspaces of V,...
Let V be an n-dimensional vector space. Let W and W2 be unequal subspaces of V, each of dimension n - 1. Prove that V =W1 + W2 and dim(Win W2) = n - 2.
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 V be an n-dimensional vector space and W a vector space that is isomorphic to...
Let V be an n-dimensional vector space and W a vector space that is isomorphic to V. Prove that W is also n-dimensional. Give a clear, detailed, step-by-step argument using the definitions of "dimension" and "isomorphic" the Definiton of isomorphic:  Let V be an n-dimensional vector space and W a vector space that is isomorphic to V. Prove that W is also n-dimensional. Give a clear, detailed, step-by-step argument using the definitions of "dimension" and "isomorphic" The Definition of dimenion: the...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT