A field is a triple \((K, +, \boldsymbol{\cdot})\), where \(K\) is a set and \(+, \boldsymbol{\cdot}\) are maps (or in detail, binary operations) \(K \times K \longrightarrow K\), satisfying the following axioms [1], [2]:
For the addition operation \(+\):
For the multiplication operation \(\boldsymbol{\cdot}\):
Distributivity of multiplication over addition
Some remarks regarding to fields
Let \((K,+,\boldsymbol{\cdot})\) be a field. a \(K\)-vector space — a vector space over field \(K\) — is a triple \((V, \oplus, \odot)\), where V is a set and \(\oplus, \odot\) are binary operations called addition and s-multiplication, respectively [1]–[3]:
\[ \begin{align*} \oplus &: V \times V \longrightarrow V \\ \odot &: K \times V \longrightarrow V \end{align*} \]
satisfying the following axioms:
Again, some remarks regarding to K-vector spaces
We define a set \(P_n(\mathbb{R})\), where \(P_n(\mathbb{R})\) is a set of \(n\)-th order polynomials with real coefficients: \[ P_n(\mathbb{R}) = \bigg\{ p: \mathbb{R} \rightarrow \mathbb{R} \; | \; p(x)=\sum_{n\in N}a_nx^n, \;\; a_n \in \mathbb{R}, \;\; N = \{ 0, 1, 2, \cdots n \} \bigg\} \] and define addition \(\oplus: P_n(\mathbb{R}) \times P_n(\mathbb{R}) \longrightarrow P_n(\mathbb{R})\) via point-wise operation:
\[ \forall x \in \mathbb{R}, \forall p_1, p_2 \in P_n(\mathbb{R}): \;\; (p_1 \oplus p_2) (x) = p_1(x) + p_2(x) \] and s-multiplication \(\odot:\mathbb{R} \times P_n(\mathbb{R}) \longrightarrow P_n(\mathbb{R})\) over field \(\mathbb{R}\). \[ \forall x, \lambda \in R, \forall p \in P_n(\mathbb{R}): (\lambda \odot p)(x) = \lambda p(x) \] Then it is straightforward to show that \((P_n(\mathbb{R}), \oplus, \odot)\) is a vector space.
For any positive integer \(n\), the set of all \(n\)-tuples of elements of field \((K,+,\cdot)\) forms a vector space over \(K\). We often call this vector space as coordinate space and denote it as \(K^{n}\). The formal definition is:
\[ K^{n} = \big\{ (k_1, k_2, \cdots, k_n) \; | \; k_1, k_2, \cdots, k_n \in K \big\} \] and define addition \(\oplus: K^n \times K^n \longrightarrow K^n\) via point-wise operation:
\[ \forall a, b \in K^n: \;\; a \oplus b = (a_1, a_2, \cdots a_n) \oplus (b_1, b_2,\dots, b_n) = (a_1+b_1, a_2+b_2, \cdots, a_n+ b_n) \] and s-multiplication \(\odot:\mathbb{R} \times P_n(\mathbb{R}) \longrightarrow P_n(\mathbb{R})\) over field \(\mathbb{R}\). \[ \lambda \in K, \forall a \in K^n: \lambda \odot a = \lambda \odot (a_1, a_2, \cdots, a_n) = (\lambda \cdot a_1, \lambda \cdot a_2, \cdots, \lambda \cdot a_n ) \]
Again, it is straightforward (but labor-intensive) to show that \(K^n, \oplus, \odot)\) is a vector space. Just for an example, we show when field \(K\) is \((\mathbb{R}, +, \cdot)\) and \(n=3\).
This example is directly from this video. Warning! The proof is labor-intensive.
Let \(V=\mathbb{R}^3\) be a set of all real triples. We equip the set \(V\) with addition \(\oplus: V \times V \longrightarrow V\) and s-multiplication \(\odot: \mathbb{R} \times V \longrightarrow V\), defined by: \[ (a,b,c)\oplus (d,e,f) \triangleq (a+d, b+e, c+f) \] and \(\forall \lambda \in \mathbb{R}\): \[ \lambda \odot (a,b,c) \triangleq (\lambda a, \lambda b, \lambda c) \] Our goal is to check \((V,\oplus, \odot)\) is a vector space.
\[ (a,b,c) \oplus (d,e,f) = (a+d, b+e, c+f)=(d+a, e+b, f+c) =(d,e,f) \oplus (a,b,c) \]
\[ \begin{align*} \big((a,b,c) \oplus (d,e,f)\big) \oplus (g,h,i) &= (a+d, b+e, c+f) \oplus (g,h,i) = (a+d+g, b+e+h, c+f+i) \\ &= (a,b,c) \oplus (d+g, e+h, f+i) = (a,b,c) \oplus \big((d,e,f) \oplus (g,h,i)\big) \end{align*} \]
\[ \exists n =(0,0,0) \in V: (0,0,0)+(a,b,c) = (a,b,c) \]
\[ \forall v = (a,b,c) \in V: \exists (-v)\triangleq(-a,-b,-c): \;\; v\oplus (-v) = n \]
\[ \forall \lambda, \mu \in \mathbb{R}: \;\; (\lambda \cdot \mu) \odot (a,b,c) = (\lambda\mu a, \lambda\mu b, \lambda\mu c) = \lambda \odot (\mu a, \mu b, \mu c) = \lambda \odot ( \mu \odot (a,b,c)) \]
\[ \begin{align*} \forall \mu \in \mathbb{R}: \;\; \mu \odot \big((a,b,c)\oplus (d,e,f)\big) &= \mu \odot (a+d, b+e, c+f) = (\mu a + \mu d, \mu b + \mu e, \mu c + \mu f) \\ &= (\mu a, \mu b, \mu c) \oplus (\mu d, \mu e, \mu f) = (\mu \odot (a,b,c)) \oplus (\mu \odot (d,e,f)) \end{align*} \]
\[ \begin{align*} \forall \mu, \lambda \in \mathbb{R}: \;\; (\mu + \lambda) \odot (a,b,c) &= \big( (\mu+\lambda)a, (\mu+\lambda)b, (\mu+\lambda)c \big) = (\mu a + \lambda a, \mu b + \lambda b, \mu c + \lambda c)\\ &= (\mu a, \mu b, \mu c ) \oplus (\lambda a, \lambda b, \lambda c) = ( \mu \odot (a,b,c) ) \oplus ( \lambda \odot (a,b,c) ) \end{align*} \]
\[ \begin{align*} 1 \odot (a,b,c) = (a,b,c) \end{align*} \]
Since \((V,\oplus, \odot)\) satisfies all 8 axioms, \((V,\oplus, \odot)\) is a vector space.