The Vector Space of n-Component Vectors

Table of Contents

From the Vector Spaces page, recall 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).

We will now verify the set of n-component vectors or "lists" as a vector space.

Let $\mathbb{F}^n$ be the set of n-component vectors whose components come from either the set of real numbers or complex numbers. Let $u, v, w \in \mathbb{F}^n$ such that $$u = (u_1, u_2, ..., u_n)$$ , $$v = (v_1, v_2, ..., v_n$$ and $$w = (w_1, w_2, ..., w_n)$$ , and define addition to be standard vector addition, and scalar multiplication to be standard scalar multiplication. Let $a, b \in \mathbb{F}$ be scalars from the field of real numbers of complex numbers.

We will now verify all ten axioms to show that $\mathbb{F}^n$ is a vector space.

1. $$u + v = (u_1 + v_1, u_2 + v_2, ..., u_n + v_n) = (v_1 + u_1, v_2 + u_2, ..., v_n + u_n) = v + u$$ .

2. $$u + (v + w) = (u_1 + [v_1 + w_1], u_2 + [v_2 + w_2], ..., u_n + [v_n + w_n]) = ([u_1 + v_1] + w_1, [u_2 + v_2] + w_2, ..., [u_n + v_n] + w_n) = (u + v) + w$$ .

3. Let $0 = \underbrace{(0, 0, ..., 0)}_{\mathrm{n-times}}$ . Then $$u + 0 = (u_1, u_2, ..., u_n) + (0, 0, ..., 0) = (u_1 + 0, u_2 + 0, ..., u_n + 0) = u$$ .

4. The additive inverse of $$u$$ is $$-u = (-u_1, -u_2, ..., -u_n)$$ and so $$u + (-u) = (u_1 - u_1, u_2 - u_2, ..., u_n - u_n) = (0, 0, ..., 0) = 0$$ .

5. $$k(lu) = a(bu_1, bu_2, ..., bu_n) = (abu_1, abu_2, ..., abu_n) = ([ab]u_1, [ab]u_2, ..., [ab]u_n) = (ab)(u_1, u_2, ..., u_n) = (ab)u$$ .

6. The scalar $$1$$ is a multiplicative identity, that is $$1u = 1(u_1, u_2, ..., u_n) = u$$ .

7. $$a(u + v) = a(u_1 + v_1, u_2 + v_2, ..., u_n + v_n) = (au_1 + av_1, au_2 + av_2, ..., au_n + av_n) = (au_1, au_2, ..., au_n) + (av_1, av_2, ..., av_n) = au + av$$ .

(1)

\begin{align} \quad (a+b)u = ([a+b]u_1, [a+b]u_2, ..., [a+b]u_n) = (au_1 + bu_1, au_2 + bu_2, ..., au_n + bu_n) \\ = (au_1, au_2, ..., au_n) + (bu_1, bu_2, ..., bu_n) = au + bu \end{align}

9. Since $$u + v = (u_1 + v_1, u_2 + v_2, ..., u_n + v_n)$$ , we have that $(u + v) \in \mathbb{F}^n$ .

10. Since $$au = (au_1, au_2, ..., au_n)$$ we have that $(au) \in \mathbb{F}^n$ .

Since $\mathbb{F}^n$ under the defined operations of addition and multiplication satisfies all ten axioms of a vector space, then we have that $\mathbb{F}^n$ is in fact a vector space.

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The Vector Space of n-Component Vectors