Question

Suppose that {v1, v2, u} and {v1, v2, w}, are both bases for some subspace V...

Suppose that {v1, v2, u} and {v1, v2, w}, are both bases for some subspace V ⊆ R n .

(a) Demonstrate with an example that it is possible for {u, w} to be linearly independent.

(b) Assuming now that {u, w} is linearly independent, decide whether {v1, u, w} is necessarily a basis for V , giving justification for your answer.

For both parts it isn’t sufficient just to give answers, you must give convincing arguments, making appropriate use of mathematical notation and terminology to support your answers. In particular if you claim something is or isn’t linearly independent, or a basis, you must show it using either the definition or results from class, unsupported claims will not be accepted, you will be marked on how well you make your case.

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
Suppose v1, v2, . . . , vn is linearly independent in V and w ∈...
Suppose v1, v2, . . . , vn is linearly independent in V and w ∈ V . Show that v1, v2, . . . , vn, w is linearly independent if and only if w ∈/ Span(v1, v2, . . . , vn).
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=[1,0,10], v2=[0,1,0,1] and let W be the subspace of R^4 spanned by v1 and v2....
let v1=[1,0,10], v2=[0,1,0,1] and let W be the subspace of R^4 spanned by v1 and v2. A. convert {v1,v2} into an orhonormal basis of W. Basis = B.find the projection of b=[-1,-2,-2,-1] onto W C.find two linear independent vectors in R^4 perpendicular to W. vectors =
Let W be a subspace of R^4 spanned by v1 = [1,1,2,0] and v2 = 2,-1,0,4]....
Let W be a subspace of R^4 spanned by v1 = [1,1,2,0] and v2 = 2,-1,0,4]. Find a basis for W^T = {v is in R^2 : w*v = 0 for w inside of W}
Suppose ⃗v1,⃗v2,⃗v3,⃗v4 ∈ R3. Let V = {⃗v1,⃗v2,⃗v3,⃗v4} and let X = [⃗v1|⃗v2|⃗v3|⃗v4] be the matrix...
Suppose ⃗v1,⃗v2,⃗v3,⃗v4 ∈ R3. Let V = {⃗v1,⃗v2,⃗v3,⃗v4} and let X = [⃗v1|⃗v2|⃗v3|⃗v4] be the matrix whose columns are ⃗v1,⃗v2,⃗v3,⃗v4. Suppose further that every subset Y ⊂ V of size two is linearly independent. Explain what form(s) rref(X), the reduced row echelon form of X, must take in this case. Hint: you won’t be able to pin down exact numbers for every entry of rref(X), but you might know things like whether the entry can be zero or not, etc.
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 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 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 a three-dimensional vector space with ordered basis B = {u, v, w}. Suppose...
Let V be a three-dimensional vector space with ordered basis B = {u, v, w}. Suppose that T is a linear transformation from V to itself and T(u) = u + v, T(v) = u, T(w) = v. 1. Find the matrix of T relative to the ordered basis B. 2. A typical element of V looks like au + bv + cw, where a, b and c are scalars. Find T(au + bv + cw). Now that you know...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT