This post expands on Harmonic Algebra posted *here*.

A vector space, or better a *vector realm*, to avoid connecting it with physical space, is a set V on which two operations + and · are defined, called vector addition and scalar multiplication.

The operation + (vector addition) must satisfy the following conditions:

Closure: If **u** and **v** are any vectors in V, then the sum **u** + **v** belongs to V.

(1) Commutative law: For all vectors **u** and **v** in V, **u** + **v** = **v** + **u**

(2) Associative law: For all vectors **u**, **v**, **w** in V, **u** + (**v** + **w**) = (**u** + **v**) + **w**

(3) Additive identity: The set V contains an additive identity element, usually denoted by 0, such that for any vector **v** in V, 0 + **v** = **v** and **v** + 0 = **v**.

(4) Additive inverses: For any vector **v** in V, the equations **v** + **x** = 0 and **x** + **v** = 0 have a solution **x** in V, called an additive inverse of **v**, and denoted by −**v**.

The operation · (scalar multiplication) is defined between real numbers (or scalars) and vectors, and must satisfy the following conditions:

Closure: If **v** is any vector in V, and *c* is any real number, then the product *c* · **v **belongs to V.

(5) Distributive law: For all real numbers *c* and all vectors **u**, **v** in V, *c* · (**u** + **v**) = c · **u** + c · **v**

(6) Distributive law: For all real numbers *c*, *d* and all vectors **v** in V, (*c *+ *d*) · **v** = *c* · **v** + *d* · **v**

(7) Associative law: For all real numbers *c*, *d* and all vectors **v** in V, *c* · (*d* · **v**) = (*cd*) · **v**

(8) Multiplicative identity: The set V contains a multiplicative identity element, usually denoted by 1, such that for any vector **v** in V, 1 · **v** = **v**

Consider the non-zero real numbers together with the element ∞ as components of Euclidean vectors, with · as the usual dot product, and vector addition defined as harmonic addition:

which is undefined for zero vectors, but has the additive identity ∞ (infinity). It is isomorphic to the vector space (or realm) with 0 as the additive identity.

The independent variable is usually in the denominator but if the independent variable is in the numerator, then the denominator contains a dependent variable. See *here* for what this looks like. The addition of quotients with a dependent vector in the denominator follows the above.