Vector Spaces
Table of Contents

Vector Spaces

Before we define exactly what a vector space is, it is important to note that was we describe as "vectors" will be a little different than in previous pages. We will denote an object $\mathbf{u}$ in boldface and only consider it a vector under the definition of a vector space below.

Definition: A nonempty set $V$ is considered to be a Vector Space if the two operations: 1. addition of the objects $\mathbf{u}$ and $\mathbf{v}$ that produces the sum $\mathbf{u} + \mathbf{v}$, and, 2. multiplication of these objects $\mathbf{u}$ with a scalar $a$ that produces the product $a \mathbf{u}$, are both defined and the ten axioms below hold. Furthermore, if $V$ is a vector space then the objects in $V$ are called vectors:

1. $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$ (Commutativity of vector addition).
2. $\mathbf{u} + (\mathbf{v} + \mathbf{w}) = (\mathbf{u} + \mathbf{v}) + \mathbf{w}$ (Associativity of vector addition).
3. There exists a zero vector $\mathbf{0}$ such that $\mathbf{0} + \mathbf{u} = \mathbf{u} + \mathbf{0} = \mathbf{u}$ (Existence of an additive identity).
4. For every $\mathbf{u} \in V$, there exists a vector $-\mathbf{u}$ such that $\mathbf{u} + (-\mathbf{u}) = (-\mathbf{u}) + \mathbf{u} = \mathbf{0}$ (Existence of an additive inverses).
5. $a(b\mathbf{u}) = (ab)\mathbf{u}$. (Associativity of scalar multiplication)
6. $1\mathbf{u} = \mathbf{u}$ (Existence of a multiplicative identity).
7. $a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}$ (Distributivity of a scalar multiplication over vector addition).
8. $(a + b)\mathbf{u} = a\mathbf{u} + b\mathbf{u}$. (Distributivity of scalar multiplication over field addition)
9. If $\mathbf{u}, \mathbf{v} \in V$, then $(\mathbf{u} + \mathbf{v}) \in V$ (Closure under addition).
10. If $a$ is any scalar and $\mathbf{u} \in V$, then $a\mathbf{u} \in V$ (Closure under scalar multiplication).

We note that from this definition, any sort of "object" can become a vector if the set of objects $V$ satisfies all parts of the definition. Furthermore, it should be noted that the field to which we define a vector space over is also important.

Definition: A Real Vector Space is a vector space over the field of real numbers $\mathbb{R}$. A Complex Vector Space is a vector space over the field of complex numbers $\mathbb{C}$.

In general, we will say that a vector space is an $\mathbb{F}$ vector space (a vector space over the field $\mathbb{F}$) when it is not important to specify the field.

We will now look at some examples of vector spaces and prove they are in fact vector spaces by verifying that the axioms hold under some defined operations of vector addition and scalar multiplication.

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