For x, y in a vector space V, c in F, if cx=0 then c=0?

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Homework Help Overview

The discussion revolves around a property in vector spaces, specifically examining the condition under which the scalar multiplication of a vector results in the zero vector. The original poster questions whether, given that \( cx = 0 \), it necessarily follows that \( c = 0 \), with \( x \) being a vector in a vector space \( V \) and \( c \) a scalar in a field \( F \).

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of the statement, particularly questioning whether the vector \( x \) must be non-zero for the proposition to hold. There is mention of using the cancellation law and the uniqueness of the zero vector in the discussion.

Discussion Status

Some participants have provided guidance on how to approach the proof, suggesting starting from the assumption that \( c \neq 0 \) and exploring the consequences. The discussion appears to be productive, with attempts to clarify the conditions under which the original statement is valid.

Contextual Notes

There is an implicit assumption that \( x \) is non-zero for the proposition to be considered, which some participants have noted. The original poster references a specific source for the problem, indicating a formal context for the discussion.

julypraise
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Homework Statement



For x, y in a vector space V, c in F, if cx=0 then c=0?
How do you prove this?
This is originally from Friedman, Linear Algebra, p.12.
To prove this, I can use a few facts:

(1) cancellation law
(2) 0, -x are unique
(3) 0x = 0

with basic vector space definition.

Homework Equations





The Attempt at a Solution

 
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Welcome to PF, julypraise! :smile:

julypraise said:

Homework Statement



For x, y in a vector space V, c in F, if cx=0 then c=0?
How do you prove this?
This is originally from Friedman, Linear Algebra, p.12.
To prove this, I can use a few facts:

(1) cancellation law
(2) 0, -x are unique
(3) 0x = 0

with basic vector space definition.

Your proposition is only true for non-zero x.
Should I assume that is an additional constraint?

If so, then to prove it, I suggest the following approach.

Start with: suppose that c≠0, and then multiply with the inverse of c.
 
Okay, thanks. Let me follow your instruction.

Prop. For x in V not equal to 0, c in F, if cx=0 then c=0.

Proof. Suppose c is not equal 0. Then if we times 1/c on the both sides of cx=0
we get 1x = c0 = 0 Therefore x = 0, which is contradiction. QED

Okay it works fairly well. Thx.
 
Good!

Actually you get: c-1cx=c-10, therefore 1x=x=0. This is use of the cancellation law.
 

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