Question

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

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 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...
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}...
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...
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}
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?
Let V be a vector space and let v1,v2,...,vn be elements of V . Let W...
Let V be a vector space and let v1,v2,...,vn be elements of V . Let W = span(v1,...,vn). Assume v ∈ V and ˆ v ∈ V but v / ∈ W and ˆ v / ∈ W. Define W1 = span(v1,...,vn,v) and W2 = span(v1,...,vn, ˆ v). Prove that either W1 = W2 or W1 ∩W2 = W.
† Let β={v1,v2,…,vn} be a basis for a vector space V and T:V→V be a linear...
† Let β={v1,v2,…,vn} be a basis for a vector space V and T:V→V be a linear transformation. Prove that [T]β is upper triangular if and only if T(vj)∈span({v1,v2,…,vj}) j=1,2,…,n. Visit goo.gl/k9ZrQb for a solution.
1. Let v1,…,vn be a basis of a vector space V. Show that (a) for any...
1. Let v1,…,vn be a basis of a vector space V. Show that (a) for any non-zero λ1,…,λn∈R, λ1v1,…,λnvn is also a basis of V. (b) Let ui=v1+⋯+vi, 1≤i≤n. Show that u1,…,un is a basis of V.
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.
Definition. Let S ⊂ V be a subset of a vector space. The span of S,...
Definition. Let S ⊂ V be a subset of a vector space. The span of S, span(S), is the set of all finite linear combinations of vectors in S. In set notation, span(S) = {v ∈ V : there exist v1, . . . , vk ∈ S and a1, . . . , ak ∈ F such that v = a1v1 + . . . + akvk} . Note that this generalizes the notion of the span of a...