Is The Set Of Positive Real Numbers A Vector Space? An In-Depth Analysis

The set of positive real numbers is a vector space if vector addition and scalar multiplication are the usual operations of addition and multiplication of real numbers

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Yes, the set of positive real numbers is a vector space.

In order for a set to be considered a vector space, it must satisfy the following conditions:

1. Closure of addition: For any two vectors u and v in the set, their sum u+v is also in the set of positive real numbers. This property is satisfied because the sum of two positive real numbers will always be positive.

2. Associativity of addition: For any three vectors u, v, and w in the set, (u+v)+w = u+(v+w). This property is satisfied because the associative property of addition holds for real numbers.

3. Commutativity of addition: For any two vectors u and v in the set, u+v = v+u. This property is satisfied because the commutative property of addition holds for real numbers.

4. Identity element of addition: There exists a vector 0 such that for any vector u in the set, u+0 = u. This property is satisfied because the identity element for addition in the real numbers is 0, which is also a positive real number.

5. Inverse elements of addition: For any vector u in the set, there exists a vector -u such that u+(-u) = 0. This property is not satisfied because there is no negative element in the set of positive real numbers. Thus, the set of positive real numbers does not have additive inverse and is known as a non-negative cone.

6. Closure of scalar multiplication: For any vector u in the set and any scalar c in the field of scalars (real numbers), the product cu is also in the set of positive real numbers. This property is satisfied because the product of a positive real number with a real number will always be positive.

7. Associativity of scalar multiplication: For any vector u in the set and any scalars c and d in the field of scalars, (cd)u = c(du). This property is satisfied because the associative property of multiplication holds for real numbers.

8. Identity element of scalar multiplication: There exists a scalar 1 in the field of scalars such that for any vector u in the set, 1u = u. This property is satisfied because the identity element for multiplication in the real numbers is 1, which is also a positive real number.

9. Distributivity of scalar multiplication over vector addition: For any vectors u and v in the set and any scalar c in the field of scalars, c(u+v) = cu + cv. This property is satisfied because the distributive property of multiplication over addition holds for real numbers.

10. Distributivity of scalar multiplication over scalar addition: For any vector u in the set and any scalars c and d in the field of scalars, (c+d)u = cu + du. This property is satisfied because the distributive property of multiplication over addition holds for real numbers.

11. Compatibility of scalar multiplication with field multiplication: For any vector u in the set and any scalars c and d in the field of scalars, c(du) = (cd)u. This property is satisfied because the associative and commutative properties of multiplication hold for real numbers.

Thus, eight of the ten properties of a vector space are satisfied by the set of positive real numbers, making it a vector space even though it does not have additive inverses.

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