Vector spaces
Definition 1: a vector space \(X\) over a scalar field \(F\) is a non-empty set, on which two algebraic operations are defined; vector addition and scalar multiplication. Such that
- \((X, +)\) is a commutative group with neutral element 0.
- the scalar multiplication satisfies \(\forall x, y \in X\) and \(\lambda, \mu \in F\)
- \(\lambda (x + y) = \lambda x + \lambda y\),
- \((\lambda + \mu) x = \lambda x + \mu x\),
- \(\lambda (\mu x) = (\lambda \mu) x\),
- \(1 x = x\).
When \(F = \mathbb{R}\) we have a real vector space while when \(F = \mathbb{C}\) we have a complex vector space.
We have that the metric spaces \(\mathbb{R}^n\), \(C\), \(l^p\) and \(l^\infty\) are also vector spaces.
Proof:
I am too lazy to add this trivial proof. Maybe some time in the future, if I do not forget.
Definition 2: a subspace of a vector space \(X\) is a non-empty subset \(M\) of \(X\), such that \(\forall x, y \in M\) and \(\lambda, \mu \in F\):
\[ \lambda x + \mu y \in M, \]with \(M\) itself a vector space.
A special subspace \(M\) of a vector space \(X\) is the improper subspace \(M = X\). Every other subspace of \(X\) is a proper subspace.
Linear combinations
Definition 3: a linear combination of the vectors \(\{x_i\}_{i=1}^n\) with \(n \in \mathbb{N}\) is vector of the form
\[ \alpha_1 x_1 + \dots + \alpha_n x_n = \sum_{i=1}^n \alpha_i x_i, \]with \(\{\alpha_i\}_{i=1}^n \in F\).
The set of all linear combinations of a set of vectors is defined as follows.
Definition 4: the span of a subset \(M \subset X\) of a vector space \(X\), denoted by \(\mathrm{span}(M)\), is the set of all linear combinations of vectors from \(M\).
It follows that \(\mathrm{span}(M)\) is a subspace of \(X\).
Linear independence
Definition 5: a finite subset of vectors \(M = \{x_i\}_{i=1}^n\) is linearly independent if
\[ \sum_{i=1}^n \alpha_i x_i = 0 \implies \forall i \in \{1, \dots, n\}: \alpha_i = 0. \]
The converse may also be defined.
Definition 6: a finite subset of vectors \(M = \{x_i\}_{i=1}^n\) is linearly dependent if \(\exists \{\alpha_i\}_{i=1}^n \in F\) not all zero such that
\[ \sum_{i=1}^n \alpha_i x_i = 0. \]
The notions of linear dependence and independence may also be extended to infinite subsets.
Definition 7: a subset \(M\) of a vector space \(X\) is linearly independent if every non-empty finite subset of \(M\) is linearly independent.
While the converse in this case is defined by the contradiction.
Definition 8: a subset \(M\) of a vector space \(X\) is linearly dependent if \(M\) is not linearly independent.
Dimension and basis
Definition 9: a vector space \(X\) is finite dimensional if there exists a \(n \in \mathbb{N}\), such that \(X\) contains a set of \(n\) linearly independent vectors, while every set of \(n+1\) vectors in \(X\) is linearly dependent. In this case \(n\) is the dimension of \(X\), denoted by \(\dim X = n\).
By definition \(X = \{0\}\) is finite dimensional and \(\dim X = 0\).
Definition 10: if a vector space \(X\) is not finite dimensional then \(X\) is infinite dimensional.
The following definition of a basis is both relevant to finite and infinite dimensional vector spaces.
Definition 11: a basis \(B\) of a vector space \(X\) is a linearly independent subset of \(X\), that spans \(X\).
Such a set \(B\) is also called a Hamel basis of \(X\).
Theorem 1: every vector space \(X\) has a Hamel basis.
Proof:
Read it again, a proof is not necessary.
Theorem 2: let \(X\) be a vector space with \(\dim X = n \in \mathbb{N}\). Then any proper subspace \(M \subset X\) has dimension less than \(n\).
Proof:
If \(n = 0\), then \(X = \{0\}\) and \(X\) has no proper subspace.
If \(\dim M = 0\), then \(M = \{0\}\) and \(X \neq M \implies \dim X \geq 1\).
If \(\dim M = n\) then \(M\) would have a basis of \(n\) elements, which would also be a basis for \(X\) since \(\dim X = n\), so that \(X = M\).
This shows that any linearly independent set of vectors in \(M\) must have fewer than \(n\) elements and \(\dim M < n\).