The Zero Vector Space

Recall from the Vector Spaces page the definition of a Vector Space:

Definition: A nonempty set $V$ is considered 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).

The simplest vector space that exists is simply the zero vector space, that is the set $\{ 0 \}$ whose only element is $0$ combined with the operations of standard addition and standard scalar multiplication. We will verify that all ten axioms hold for this vector space, much of which is redundant. Let $a, b, \in \mathbb{F}$.

• 1. $0 + 0 = 0$

• 2. $0 + (0 + 0) = (0 + 0) + 0$.

• 3. The zero vector is $0$, that is $0 + 0 = 0$.

• 4. The additive inverse of $0$ is $-0$, that is $0 + (-0) = 0$.

• 5. $a(b0) = (ab)0 = 0$.

• 6. Any scalar $a \in \mathbb{F}$ works as a multiplicative identity, that is $a0 = 0$.

• 7. $a(0 + 0) = a0 + a0 = 0$.

• 8. $(a + b)0 = a0 + b0 = 0$.

• 9. Since $0 + 0 = 0$, we note that $(0 + 0) \in \{ 0 \}$ and so $\{ 0 \}$ is closed under addition.

• 10. Since $a0 = 0$, we note that $(a0) \in \{ 0 \}$ and so $\{ 0 \}$ is closed under scalar multiplication.

Therefore the set $\{ 0 \}$ containing the operations of standard addition and standard multiplication is a vector space since all ten vector space axioms hold.