A community for students.
Here's the question you clicked on:
 0 viewing
 3 years ago
Let v1, v2, ..., Vn be a list of nonzero vectors in a vector space V such that no vector in the list is a linear combination of its predecessors. Show that the vectors in the list form an independent set.
 3 years ago
Let v1, v2, ..., Vn be a list of nonzero vectors in a vector space V such that no vector in the list is a linear combination of its predecessors. Show that the vectors in the list form an independent set.

This Question is Closed

rulnick
 3 years ago
Best ResponseYou've already chosen the best response.1It's almost selfevident from the definition of independence, but here goes. A = no vector is a linear comb of its predecessors B = vectors are indep We will show A => B by showing not B => not A. Assume the vectors are not indep. Then there exists at least one vector that can be expressed as a linear combination of others. Call it v*. If all of the others in the linear combination are predecessors of v* then we are done, so assume not: assume at least one vector in the linear combination follows v*. Then this vector is a linear combination of the others together with v*, which contradicts A. Hence not B => not A, so A => B.

brinethery
 3 years ago
Best ResponseYou've already chosen the best response.3I just read this twice, and I'll have to really read these definitions and proofs carefully. Thank you for your detailed answer, I appreciate it.

rulnick
 3 years ago
Best ResponseYou've already chosen the best response.1No problem. It may help to think of this in the case of just three vectors. For example ...

rulnick
 3 years ago
Best ResponseYou've already chosen the best response.1If you have v1, v2, and v3, and they are dependent, then a v1 + b v2 + c v3 = 0. Which means a v1 + c v3 = b v2 ( v2 is a lin comb, but not of its predecessors) but then a v1 + b v2 = c v3 (so v3 *is* a lin comb of its predecessors).
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.