Prove vector space postulate 1.X = X is independent of others

In summary, the conversation discusses the vector space postulate 1.X = X and how it cannot be derived from the other postulates. The only hint given is to construct a "pseudo-scalar product" and the conversation goes on to show how this can be done. The conversation also mentions the vector space axioms and how proving that 2d) is independent from the other axioms is the main objective. The example given involves using a zero dimensional subspace to satisfy 2a-c but not 2d). This shows that the scalar products that satisfy axioms 2a-c are projections onto some subspace of the vector space.
  • #1
xalvyn
17
0
Hi everyone,

I would like to seek help in proving that the vector space postulate 1.X = X cannot be derived from the other postulates, e.g. X + 0 = X, X + (Y + Z) = (X + Y) + Z.

The only hint I am given is to construct the "pseudo-scalar product"
c # X = the projection of c.X on a fixed line.

This problem is from Birkhoff and Maclane's "A Survey of Modern Algebra".
 
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  • #2
The vector space axioms are:

1. V is an abelian group under addition, which means:
a) (X+Y)+Z=X+(Y+Z)
b) there is a vector 0 with X+0=X
c) For each X, there is a vector -X with X+(-X)=0
d) X+Y=Y+X

2. Let F be the underlying field (eg, the real numbers). Then there is a map from F x V into V (ie, for any r in F and X in V, there is a unique vector rX in V), which satisfies:
a) r(X+Y) = rX+rY
b) (r+s)X = rX+sX
c) (rs)X = r(sX)
d) 1X = X

So what you want to do is show that 2 d) is independent from the other axioms, right? What you need to do is come up with a structure that satisfies all other axioms, but not 2d). That is, find a field F (probably the real numbers) and an abelian group V (probably Rn under vector addition), and find a new definition scalar product such that 2 a-c are satisfied, but not 2). It sounds like they already did the hard part by giving you such a scalar product. All you need to do is verify it has these properties.
 
  • #3
Hi StatusX,

Thanks a many. While reading your reply, I was thinking: is it possible to define this pseudo-scalar product

r # X = 0, where X is a vector in R^N, and so forms an abelian group under addition ?

In this case, we have
2a) r # (X + Y) = r # (Z) = 0 = 0 + 0 = r # X + r # Y, where Z = X + Y

b) (r + s) # X = k # X = 0 = 0 + 0 = r # X + s # X, k = r + s

c) (rs) # X = k # X = 0 = r # 0 = r # (s # X)

d) 1 # X = 0, which is not necessarily equal to X.

Thus 2a) - c) are satisfied, but not d).

really appreciate the help, thanks.
 
  • #4
Yes, that works nicely. It looks as if the scalar products that satisfy axioms 2a-c are exactly those that are projections onto some subspace of the vector space. If this is the whole space, the last axiom is satisfied, and if it is any smaller, it is not. The example they gave as a hint used a 1 dimensional subspace (a line), where as your example uses a zero dimensional subspace, and so is even simpler.
 

1. What is the first postulate of proving a vector space?

The first postulate of proving a vector space is that the addition operation for vectors must be closed. This means that when two vectors are added together, the result must also be a vector in the same space.

2. How does postulate 1.X = X being independent of others relate to vector addition?

This postulate states that the addition of a vector to itself does not depend on any other vectors in the space. This is important because it ensures that the addition operation is consistent and does not produce different results depending on the order in which vectors are added.

3. Why is it important for the addition operation to be closed in a vector space?

If the addition operation is not closed, it means that the resulting vector may not be in the same space as the original vectors. This would violate the first postulate and would not meet the requirements of a vector space.

4. How does postulate 1.X = X being independent of others relate to vector scalar multiplication?

This postulate also applies to scalar multiplication, where a scalar is multiplied by a vector. It states that the result of multiplying a vector by a scalar is still a vector in the same space, regardless of the values of the scalar or the vector.

5. What is the purpose of postulate 1.X = X being independent of others in proving a vector space?

This postulate is crucial in proving that a set of vectors and their corresponding operations meet the requirements of a vector space. By ensuring that vector addition and scalar multiplication are closed and independent of other vectors, we can show that the set satisfies the first postulate and is a valid vector space.

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