Vector Space

A vector space (also called a linear space) is a collection of objects called vectors, which can be added together and can be multiplied by scalars, where scalars are elements from a field.

More formally, a vector space $V$ over a field $F$ (such as the real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$) is a set equipped with two operations: vector addition and scalar multiplication. These operations must satisfy the following eight axioms:

1. Associativity of addition:

$$(u + v) + w = u + (v + w)\forall u, v, w \in V $$

2. Commutativity of addition:

$$u + v = v + u \forall u, v \in V$$

3. Identity element of addition: There exists an element $0 \in V$, called the zero vector

$$v + 0 = v \forall v \in V$$

4. Inverse elements of addition: For every $v$ $\in$ $V$, there exists an element $-v$ $\in$ $V$, called the additive inverse of $v$, such that $v + (-v) = 0$
5. Compatibility of scalar multiplication with field multiplication: $\forall a, b$ $\in$ $F$ and all $v$ $\in$ $V$, we have $a \cdot (b \cdot v) = (a \cdot b) \cdot v$
6. Identity element of scalar multiplication: $\forall v$ $\in$ $V$, we have $1 \cdot v = v$, where $1$ denotes the multiplicative identity $\in$ $F$
7. Distributivity of scalar multiplication with respect to vector addition: $\forall a$ $\in$ $F$ and all $u, v$ $\in$ $V$, we have $a \cdot (u + v) = a \cdot u + a \cdot v$
8. Distributivity of scalar multiplication with respect to scalar addition: $\forall a, b$ $\in$ $F$ and all $v$ $\in$ $V$, we have $(a + b) \cdot v = a \cdot v + b \cdot v$

Examples of vector spaces include the set of all real numbers (which is a vector space over itself), the set of all complex numbers (also a vector space over itself), the set of all $n$-tuples of real numbers (a vector space over the real numbers), the set of all sequences of real numbers, the set of all functions from the real numbers to the real numbers, and many others.