Prove that the additive identity in a vector space is unique

In summary, the conversation discusses how to prove that the additive identity in a vector space is unique. The solution involves assuming that it is not unique and showing that this leads to a contradiction. It is important to note that the assumption should not already include the statement that the additive identity is unique.
  • #1
zeion
466
1

Homework Statement



Prove that the additive identity in a vector space is unique


Homework Equations



Additive identity

There is an element 0 in V such that v + 0 = v for all v in V

The Attempt at a Solution



Assume that the additive identity is NOT unique, then there exists y, z belong to V such that
A + y = A + z = A, then y = z = 0, which is a contradiction.

Is this enough to prove??
 
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  • #2
zeion said:
… Assume that the additive identity is NOT unique, then there exists y, z belong to V such that
A + y = A + z = A, then y = z = 0, which is a contradiction.

Is this enough to prove??

Hi zeion! :smile:

hmm … you're assuming that A - A = 0, which is sort-of begging the question.

Hint: what is y + z ? :wink:
 
  • #3
Since y, z belong to V, and y, z are the zero vectors in V, then
y + z = y = z, which is a contradiction.
 
  • #4
Yup! :biggrin:

(except it's not actually a contradiction … unless you state at the start that y and z are different, which you don't have to).
 
  • #5
Ooh okay! Thanks ^_^

But why couldn't I say that A - A = 0?
Could I do that if I stated that I assumed A was in V?

..or is it because then I would be assuming that A was unique?
 
  • #6
… that 0 was unique? yes! :wink:
 

1. What is the definition of an additive identity in a vector space?

The additive identity in a vector space is a special element, typically denoted as 0, that when added to any vector in the space results in the vector being unchanged.

2. Why is it important to prove that the additive identity in a vector space is unique?

It is important to prove that the additive identity in a vector space is unique because it ensures that there is only one element that satisfies the definition of an additive identity. This helps to maintain consistency and coherence within the mathematical system.

3. How is the uniqueness of the additive identity proven in a vector space?

The uniqueness of the additive identity can be proven by assuming the existence of two different additive identities, say 0 and 0'. By the definition of an additive identity, both 0 and 0' would have to result in the same vector when added to any other vector in the space. This would then lead to the conclusion that 0 = 0', proving the uniqueness of the additive identity.

4. Can there be more than one additive identity in a vector space?

No, there can only be one additive identity in a vector space. If there were multiple elements that satisfied the definition of an additive identity, it would lead to inconsistencies and contradictions in the mathematical system.

5. Is the uniqueness of the additive identity a universal property of all vector spaces?

Yes, the uniqueness of the additive identity is a universal property of all vector spaces. This means that it holds true for all vector spaces, regardless of their specific properties or dimensions.

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