Question

proof of the following statement If S= {v1, v2, ...., vr} is a nonempty set of...

proof of the following statement

If S= {v1, v2, ...., vr} is a nonempty set of vectors in a vector space V then : W is the smallest subspace of V that contains all of the vectors in S in the sense that any other subspace of V that contains those vectors must contain W.

Math   Algebra   
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...
1. Prove that if {⃗v1, ⃗v2, ⃗v3} is a linear dependent set of vectors in V...
1. Prove that if {⃗v1, ⃗v2, ⃗v3} is a linear dependent set of vectors in V , and if ⃗v4 ∈ V , then {⃗v1, ⃗v2, ⃗v3, ⃗v4} is also a linear dependent set of vectors in V . 2. Prove that if {⃗v1,⃗v2,...,⃗vr} is a linear dependent set of vectors in V, and if⃗ vr + 1 ,⃗vr+2,...,⃗vn ∈V, then {⃗v1,⃗v2,...,⃗vn} is also a linear dependent set of vectors in V.
Prove that Let S={v1,v2,v3} be a linearly indepedent set of vectors om a vector space V....
Prove that Let S={v1,v2,v3} be a linearly indepedent set of vectors om a vector space V. Then so are {v1},{v2},{v3},{v1,v2},{v1,v3}，{v2,v3}
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 {V1, V2,...,Vn} be a linearly independent set of vectors choosen from vector space V. Define...
Let {V1, V2,...,Vn} be a linearly independent set of vectors choosen from vector space V. Define w1=V1, w2= v1+v2, w3=v1+ v2+v3,..., wn=v1+v2+v3+...+vn. (a) Show that {w1, w2, w3...,wn} is a linearly independent set. (b) Can you include that {w1,w2,...,wn} is a basis for V? Why or why not?
Suppose the vectors v1, v2, . . . , vp span a vector space V ....
Suppose the vectors v1, v2, . . . , vp span a vector space V . (1) Show that for each i = 1, . . . , p, vi belongs to V ; (2) Show that given any vector u ∈ V , v1, v2, . . . , vp, u also span V
If v1 and v2 are linearly independent vectors in vector space V, and u1, u2, and...
If v1 and v2 are linearly independent vectors in vector space V, and u1, u2, and u3 are each a linear combination of them, prove that {u1, u2, u3} is linearly dependent. Do NOT use the theorem which states, " If S = { v 1 , v 2 , . . . , v n } is a basis for a vector space V, then every set containing more than n vectors in V is linearly dependent." Prove without...
Let X be a real vector space. Suppose {⃗v1,⃗v2,⃗v3} ⊂ X is a linearly independent set,...
Let X be a real vector space. Suppose {⃗v1,⃗v2,⃗v3} ⊂ X is a linearly independent set, and suppose {w⃗1,w⃗2,w⃗3} ⊂ X is a linearly dependent set. Define V = span{⃗v1,⃗v2,⃗v3} and W = span{w⃗1,w⃗2,w⃗3}. (a) Is there a linear transformation P : V → W such that P(⃗vi) = w⃗i for i = 1, 2, 3? (b) Is there a linear transformation Q : W → V such that Q(w⃗i) = ⃗vi for i = 1, 2, 3? Hint: the...
Let V be an inner product space. Prove that if w⃗ is orthogonal to each of...
Let V be an inner product space. Prove that if w⃗ is orthogonal to each of the vectors in the set S = {⃗v1, ⃗v2, . . . , ⃗vm}, then w⃗ is also orthogonal to each of the vectors in the subspace W = SpanS of V .
Problem 6.4 What does it mean to say that a set of vectors {v1, v2, ....
Problem 6.4 What does it mean to say that a set of vectors {v1, v2, . . . , vn} is linearly dependent? Given the following vectors show that {v1, v2, v3, v4} is linearly dependent. Is it possible to express v4 as a linear combination of the other vectors? If so, do this. If not, explain why not. What about the vector v3? Anthony, Martin. Linear Algebra: Concepts and Methods (p. 206). Cambridge University Press. Kindle Edition.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT