Prove That If rv = 0 Then Either r=0 or v=0

Click For Summary

Discussion Overview

The discussion revolves around proving the statement that if the product of a scalar \( r \) and a vector \( v \) in a vector space \( V \) equals the zero vector, then either \( r \) must be zero or \( v \) must be the zero vector. The scope includes mathematical reasoning and conceptual clarification regarding vector spaces and scalar multiplication.

Discussion Character

  • Mathematical reasoning
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how to prove the statement and expresses uncertainty about the proof's validity.
  • Another participant provides a proof using a specific vector representation and concludes that either \( r = 0 \) or all components of \( v \) are zero, suggesting this proof may not be unique.
  • A different participant challenges the proof by stating it relies on choosing a basis and questions its validity across different bases.
  • In response, another participant offers a basis-free proof, arguing that if \( r \neq 0 \), multiplying both sides of \( rv = 0 \) by \( 1/r \) leads to \( v = 0 \).
  • One participant critiques the previous argument about changing bases, asserting that the origin in a vector space is not fixed.
  • Another participant adds that the result holds true for all vector spaces, regardless of dimensionality or the need for the axiom of choice in certain cases.

Areas of Agreement / Disagreement

Participants express differing views on the validity of proofs based on basis selection, with some arguing for the necessity of a basis-free approach. The discussion remains unresolved regarding the implications of changing bases and the generality of the proofs presented.

Contextual Notes

Some arguments depend on assumptions about the nature of vector spaces and the fixedness of the origin, which may not be universally applicable. The discussion also touches on the implications of dimensionality and the axiom of choice in relation to vector spaces.

broegger
Messages
257
Reaction score
0
how do you prove that if v is an element of V (a vector space), and if r is a scalar and if rv = 0, then either r = 0 or v = 0... it seems obvious, but i have no idea how to prove it...
 
Physics news on Phys.org
If v is the vector (a, b, c), then the vector that results from the multiplication rv is (ra, rb, rc). If the result is equal to 0, the zero vector, then (ra, rb, rc) = (0, 0, 0). If we write this formally we get:
[tex]ra = 0[/tex]
[tex]rb = 0[/tex]
[tex]rc = 0[/tex]
The solution is that either r equals 0, or a, b, and c all equal 0. And if a, b, and c are 0 then the vector v is (0, 0, 0) which is the zero vector.

Is this proof satisfactory? There are probably a lot of ways to prove this. :smile:
 
That proof requires you to pick a basis. If I pick a different basis, do you know that it still holds?

Here's the basis free proof. Suppose rv=0, then if r is zero we are done, if not multiply rv=0 by 1/r and we see v=0.
 
matt grime said:
That proof requires you to pick a basis. If I pick a different basis, do you know that it still holds?
Yes... if you pick your basis at (A, B, C) then the vector v becomes (a - A, b - B, c - C) and the zero vector is now (A, B, C).
[tex]r(a - A) = A[/tex]
[tex]r(b - B) = B[/tex]
[tex]r(c - C) = C[/tex]
For this to be true for r <> 0, a must equal A, b must equal B and c must qual C, and thus the vector v becomes (A, B, C) which is again the zero vector.
 
No, that isn't how one does a change of basis (of a vector space: the origin isn't fixed.)
 
It's also dimension dependent. The result is true for every vector space, even those where picking a basis, never mind solving an uncountable set of linear equations requires the axiom of choice.
 

Similar threads

Undergrad Confused about small detail in rank-nullity theorem
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Proving (x,x) = 0 implies x = 0 in real vector space V
  • · Replies 21 ·
Replies
21
Views
3K
MHB Proving Non-Degeneracy of Euclidean Inner Products
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad About the definition of vector space of infinite dimension
  • · Replies 18 ·
Replies
18
Views
4K
Undergrad Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Terminologies used to describe tensor product of vector spaces
  • · Replies 7 ·
Replies
7
Views
3K
Graduate A problem in multilinear algebra
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad The vector to which a dual vector corresponds
  • · Replies 13 ·
Replies
13
Views
6K
Undergrad Mapping tensor products into a Clifford algebra
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Two ways to define operations in a vector space
  • · Replies 7 ·
Replies
7
Views
2K