Is This Set of Triples of Real Numbers a Vector Space?

In summary, to determine if a set is a vector space, one must check if it satisfies the 10 vector space axioms. In this case, the set of all triples of real numbers with the given operations satisfies 9 of the axioms, but not the axiom for distributivity of scalars. This can be seen by comparing (m + k)(x, y, z) and k(x, y, z) + m(x, y, z), which do not result in the same values. Therefore, the set is not a vector space.
  • #1
derryck1234
56
0

Homework Statement



Show whether the set is a vector space: The set of all triples of real numbers (x, y, z) with the operations:

(x, y, z) + (x', y', z') = (x + x', y + y', z + z') and k(x, y, z) = (kx, y, z)

Homework Equations



(10 vector space axioms)

The Attempt at a Solution



I can understand 9 axioms, I just want to confirm that I am doing the right thing on this one:

(m + k)(x, y, z) = ((m+k)x, y, z), which is not equal to k(x, y, z) + m(x, y, z) = (kx, y, z) + (mx, y, z) = ((m+k)x, 2y, 2z).

Is this correct working?

Thanks

Derryck
 
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  • #2
Yup.
 

FAQ: Is This Set of Triples of Real Numbers a Vector Space?

1. What is a vector space?

A vector space is a mathematical structure composed of a set of objects called vectors, which can be added together and multiplied by scalars (numbers). These operations must satisfy a set of axioms in order for the set to be considered a vector space.

2. How do you prove that a set is a vector space?

To prove that a set is a vector space, you must show that it satisfies all of the axioms of a vector space. These include closure under vector addition and scalar multiplication, associativity and commutativity of addition, existence of an additive identity and inverse, and distributivity of scalar multiplication over vector addition.

3. Can a set be a vector space if it contains non-numeric elements?

No, a vector space must contain elements that can be added together and multiplied by scalars. These elements must also satisfy the axioms of a vector space, which require them to behave like numbers.

4. What is the difference between a vector space and a subspace?

A subspace is a subset of a vector space that also satisfies all of the axioms of a vector space. This means that all of the operations defined on the larger vector space can also be performed on the elements of the subspace. A vector space, on the other hand, is a complete mathematical structure that contains all of the necessary elements and operations.

5. Can you prove that a set is a vector space using only a subset of the axioms?

No, in order for a set to be considered a vector space, it must satisfy all of the axioms. If even one axiom is not satisfied, the set cannot be considered a vector space.

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