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

In summary, if v is an element of V (a vector space) and rv = 0, then either r = 0 or v = 0. This can be proven by showing that if rv = 0, then either r = 0 or each component of v is equal to 0. This proof holds for any basis and any dimension of the vector space.
  • #1
broegger
257
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...
 
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  • #2
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:
 
  • #3
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.
 
  • #4
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.
 
  • #5
No, that isn't how one does a change of basis (of a vector space: the origin isn't fixed.)
 
  • #6
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.
 

What does "Prove That If rv = 0 Then Either r=0 or v=0" mean?

This statement is a mathematical proof that shows that if the product of two variables, r and v, is equal to 0, then either r or v must also be equal to 0. In other words, if one of the variables is 0, then the other must also be 0 in order for the product to equal 0.

Why is it important to prove this statement?

Proving this statement is important because it demonstrates the relationship between multiplication and the concept of 0 as the identity element. It also lays the foundation for more complex mathematical proofs and equations involving variables and their products.

What is the significance of the "if...then" structure in this statement?

The "if...then" structure is a conditional statement that states the necessary condition (rv = 0) for the sufficient condition (r=0 or v=0) to be true. In this case, it is necessary for the product of r and v to be 0 in order for either r or v to be 0.

Can you provide an example to illustrate this statement?

Sure, let's say r = 0 and v = 5. The product of these two variables, rv, would equal 0, satisfying the first part of the statement. Since r = 0, this also satisfies the second part of the statement, showing that either r=0 or v=0 in order for the product to equal 0.

What are some real-life applications of this statement?

This statement can be applied in various fields such as physics, engineering, and economics to solve equations and determine unknown variables. It can also be used to prove other mathematical theories and concepts. For example, it is a fundamental principle in linear algebra and matrix operations.

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