How to Prove All 8 Vector Space Axioms

You cannot prove that an unspecified set is a vector space: the claim depends on the set, the field of scalars, and the definitions of addition and scalar multiplication. What you can do is check the vector space laws for a particular candidate.

Here is the common convention: eight algebraic laws, together with the requirement that the operations are closed on the set. Some books count closure as additional axioms, so the number in a list can vary.

The eight laws to verify

Let $V$ be a set with vector addition and scalar multiplication, and let $\mathbb{F}$ be the scalar field. For all $u,v,w\in V$ and $a,b\in\mathbb{F}$, verify:

1. Addition is commutative: $u+v=v+u$.
2. Addition is associative: $(u+v)+w=u+(v+w)$.
3. There is a zero vector: Some $0_V\in V$ satisfies $v+0_V=v$ for every $v\in V$.
4. Every vector has an additive inverse: For every $v\in V$, some $-v\in V$ satisfies $v+(-v)=0_V$.
5. Scalar multiplication distributes over vector addition: $a(u+v)=au+av$.
6. Scalar addition distributes over scalar multiplication: $(a+b)v=av+bv$.
7. Scalar multiplication is compatible with field multiplication: $a(bv)=(ab)v$.
8. The scalar identity acts as the identity: $1v=v$.

Also check that $u+v\in V$ and $av\in V$ whenever $u,v\in V$ and $a\in\mathbb{F}$. These closure conditions ensure the operations actually return elements of the proposed space. The field itself is assumed to satisfy its field axioms.

Worked example: the usual vector space $\mathbb{F}^2$

Take $V=\mathbb{F}^2$, where $\mathbb{F}$ is any field. Define addition and scalar multiplication by

These operations are closed: field sums and products belong to $\mathbb{F}$, so both results are again ordered pairs in $\mathbb{F}^2$. Now check the laws for arbitrary $u=(x_1,x_2)$, $v=(y_1,y_2)$, $w=(z_1,z_2)$ and $a,b\in\mathbb{F}$.

1. Commutativity:

   $$u+v=(x_1+y_1,x_2+y_2)=(y_1+x_1,y_2+x_2)=v+u,$$

   using commutativity in $\mathbb{F}$.

2. Associativity:

   $$(u+v)+w=((x_1+y_1)+z_1,(x_2+y_2)+z_2) =(x_1+(y_1+z_1),x_2+(y_2+z_2))=u+(v+w).$$

3. Zero vector: The pair $(0,0)$ is in $V$, and

   $$(x_1,x_2)+(0,0)=(x_1,x_2).$$

   Thus $0_V=(0,0)$.

4. Additive inverse: The pair $(-x_1,-x_2)$ is in $V$, and

   $$(x_1,x_2)+(-x_1,-x_2)=(0,0).$$

5. Scalar multiplication distributes over vector addition:

   $$a(u+v)=(a(x_1+y_1),a(x_2+y_2)) =(ax_1+ay_1,ax_2+ay_2)=au+av.$$

6. Scalar addition distributes:

   $$(a+b)v=((a+b)y_1,(a+b)y_2) =(ay_1+by_1,ay_2+by_2)=av+bv.$$

7. Compatibility with scalar multiplication:

   $$a(bv)=(a(by_1),a(by_2))=((ab)y_1,(ab)y_2)=(ab)v.$$

8. Scalar identity:

   $$1v=(1y_1,1y_2)=(y_1,y_2)=v.$$

Each equality uses a field law in the coordinates, so this proves the result for every field $\mathbb{F}$, not just for real numbers.

How to handle a different candidate

Use the same checklist, but substitute the candidate’s actual operations. If the set consists of functions, for example, addition and scalar multiplication are usually defined pointwise; you then verify the laws at each input. If the operations are unusual, do not assume familiar rules still hold—test the definitions directly.

You often do not need to repeat a long calculation eight times. State that the operations are closed, identify the relevant field properties, and show enough detail that each axiom is accounted for. If one law fails for even a single allowed choice of vectors or scalars, the structure is not a vector space.

Common mistakes

• Checking only closure: Closure is necessary, but it does not imply the eight laws.
• Using the wrong zero: The zero vector is the additive identity for the defined vector addition; it is not automatically a scalar or a familiar all-zero object unless the definition makes it so.
• Skipping arbitrary choices: A proof must work for all vectors and scalars, not just a sample calculation.
• Mixing up the distributive laws: One distributes scalar multiplication across a vector sum; the other distributes scalar addition across a vector.
• Forgetting the scalar field: The proof can depend on which scalars are allowed. In particular, an expression involving division is not automatically valid over every field.