How can I prove that v and w are equal if v + x = w + x?

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Homework Help Overview

The discussion revolves around proving properties related to vector spaces, specifically focusing on the expression 0u=0 and the equality v + x = w + x. The subject area includes vector space axioms and properties of vector addition and scalar multiplication.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore various approaches to proving that 0u equals the zero vector, with some suggesting specific vector representations. Others question assumptions about vector definitions and dimensions, while one participant attempts to manipulate the expression involving v and w to derive a conclusion.

Discussion Status

The discussion is active, with participants offering different perspectives and methods for approaching the problem. Some guidance has been provided, particularly in manipulating expressions, but there is no explicit consensus on a single method or solution.

Contextual Notes

Participants are navigating the constraints of vector space definitions and the axioms governing them, with some assumptions about dimensionality and representation being questioned.

brandy
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Homework Statement


prove 0u=0 where u is in the vector space.


Homework Equations


the 10 various axioms for addition and scalar multiplication.


The Attempt at a Solution


pretty much just

(u+-u)u=0
or
(1-1)u=0
1u+-1u=0

and then i get stuck. i can prove that -1u=-u but that involves 0u-0
 
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Couldn't you let u = (a,b), then
0u
0(a,b)
(0a,0b)
(0,0)
0
 
No, Bearded Man. You made an assumption about "what vectors are" that doesn't follow from the axioms. Some vector spaces are described by components, but not all, and it's certainly not necessarily two-dimensional.

Try this:
u=1u=(1+0)u.

Distribute, and see where it takes you.
 
Here you go Brandy:

0 x u = 0 + 0 x u
= -u + u + 0 x u
= -u + (1 + 0) x u
= -u + 1 x u
= -u + u
= 0

And maybe you can help me with this one.
v + x = w + x
Prove v = w
 

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