A binary operation is a function \(\mathbb{X} \times \mathbb{X} \rightarrow \mathbb{X}\) which maps ordered pairs of elements from \(\mathbb{X}\) to other elements of \(\mathbb{X}.\)

A field \(\mathbb{F}\) is a set with two binary operations called addition and scalar multiplication. Addition \(a+b\) and scalar multiplication \(a \cdot b,\) \(a,b \in \mathbb{F}\) satisfy the field axioms:

  • Associativity of addition and scalar multiplication: \(a + (b + c) = (a + b) + c\) and \(a \cdot (b \cdot c) = (a \cdot b) \cdot c.\)
  • Commutativity of addition and scalar multiplication: \(a + b = b + a\) and \(a \cdot b = b \cdot a.\)
  • Distributivity of scalar multiplication over addition: \(a \cdot (b + c) = (a \cdot b) + (a \cdot c).\)
  • Additive and scalar multiplicative identity: \(\exists \, 0, 1 \in \mathbb{F} : a + 0 = a\) and \(a \cdot 1 = a.\)
  • Additive and scalar multiplicative inverse: \(\exists \, {-}a \in \mathbb{F} : a+(-a)=0 \,\, \forall \, a \in \mathbb{F}\) and \(\exists \, a^{-1} \in \mathbb{F} \setminus \{0\} : a \cdot a^{-1} = 1 \,\, \forall \, a \in \mathbb{F}\setminus \{0\}.\)

Usually \(\mathbb{F}=\mathbb{R}\) (real numbers) or \(\mathbb{F}=\mathbb{C}\) (complex numbers). Any element \(\lambda \in \mathbb{F}\) is a scalar.

The set \(\mathcal{V}\) over the field \(\mathbb{F}\) is a vector space if \(\mathcal{V}\) is closed under addition \(\underline{u} + \underline{v} \in \mathcal{V} \,\, \forall \, \underline{u}, \underline{v} \in \mathcal{V}\) and scalar multiplication \(\lambda\cdot \underline{u} \in \mathcal{V} \,\, \forall \, \lambda \in \mathbb{F}\) and the elements of \(\mathcal{V}\) satisfy the vector space axioms \(\forall \, \underline{u}, \underline{v}, \underline{w} \in \mathcal{V}\) and \(\forall \, \lambda, \mu \in \mathbb{F}:\)

  • Associativity of addition and scalar multiplication: \(\underline{u}+(\underline{v}+\underline{w})=(\underline{u}+\underline{v})+\underline{w}\) and \(\lambda\cdot(\mu\cdot\underline{u})=(\lambda\mu)\cdot\underline{u}.\)
  • Commutativity of addition: \(\underline{u}+\underline{v}=\underline{v}+\underline{u}.\)
  • Distributivity of scalar multiplication over addition: \((\lambda+\mu)\cdot\underline{u}=\lambda\cdot\underline{u}+\mu\cdot\underline{u}\) and \(\lambda\cdot(\underline{u}+\underline{v})=\lambda\cdot\underline{u}+\lambda\cdot\underline{v}.\)
  • Additive identity: \(\exists \, \underline{0} \in \mathcal{V} : \underline{u}+\underline{0}=\underline{u}\)
  • Scalar multiplicative identity: \(\exists \, 1 \in \mathbb{F} : 1\cdot\underline{u}=\underline{u}\)
  • Additive inverse: \(\exists \, {-}\underline{u} \in \mathcal{V} : \underline{u}+({-}\underline{u})=\underline{0}.\)

Usually \(\mathcal{V}=\mathbb{R}^n\) where \(v_1,\ldots v_n \in \mathbb{R}\) or \(\mathcal{V}=\mathbb{C}^n\) where \(v_1,\ldots v_n \in \mathbb{C}\) for the set of \(n\)-tuples \(\underline{v}=(v_1,\ldots,v_n).\) Any element \(\underline{v} \in \mathcal{V}\) is a vector.

Let \(\mathcal{V}\) be a vector space over the field \(\mathbb{F}.\) A subspace \(\mathcal{U}\) of \(\mathcal{V}\) is a subset \(\mathcal{U} \subseteq \mathcal{V}\) that is itself a vector space. \(\mathcal{U}\) can be a subset of \(\mathcal{V}\) and not a subspace if it does not satisfy all vector space axioms.

Let \(\mathcal{V}\) be a vector space over the field \(\mathbb{F}\) and \(U\) be a subset of \(\mathcal{V}.\) The vectors \(\underline{u} \in U\) are linearly dependent if at least one vector \(\underline{u}_k\) can be written as a linear combination of the other vectors. In other words, for some \(k, 1 \leq k \leq n\) there exists a set of scalars \(\alpha_1,\ldots,\alpha_{k-1}, \alpha_{k+1},\ldots,\alpha_n\) such that

\[\underline{u}_k = \sum_{i=1, i\neq k}^n \alpha_i\underline{u}_i\]

If there are no linearly dependent vectors in \(U\) then its vectors are linearly independent. Equivalently, the vectors in \(U\) are linearly independent if and only if the solution to \(\alpha_1\underline{u}_1+\ldots+\alpha_n\underline{u}_n=\underline{0}\) is the trivial solution \(\alpha_1=\ldots=\alpha_n=0.\) \(U\) is said to be linearly dependent if every finite subset of \(U\) is linearly independent.

Let \(\mathcal{V}\) be a vector space over the field \(\mathbb{F}\) and \(U\) be a subset of \(\mathcal{V}.\) \(U\) is said to span \(\mathcal{V}\) if all vectors in \(\mathcal{V}\) can be expressed as a linear combination of vectors in \(U.\) \(U\) is said to be a basis of \(\mathcal{V}\) if \(U\) spans \(\mathcal{V}\) and the vectors in \(U\) are linearly independent. The dimension of \(\mathcal{V}\) is the number of vectors in its basis. Every vector space that can be spanned has a basis and all bases for a given vector space have the same number of vectors. Hence, the dimension of a vector space is uniquely defined.