The Vector Space of n-Component Vectors
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). |
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. $a(u + v) = a(u_1 + v_1, u_2 + v_2, …, u_n + v_n) = (au_1 + av_1, au_2 + av_2, …, au_n + av_n) = (au_1, au_2, …, au_n) + (av_1, av_2, …, av_n) = au + av$.
- 8.
(1)
- 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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