4 KiB
Vector spaces
Definition 1: a vector space
X
over a scalar fieldF
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.
??? note "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 subsetM
ofX
, 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
withn \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 spaceX
, denoted by\mathrm{span}(M)
, is the set of all linear combinations of vectors fromM
.
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 spaceX
is linearly independent if every non-empty finite subset ofM
is linearly independent.
While the converse in this case is defined by the contradiction.
Definition 8: a subset
M
of a vector spaceX
is linearly dependent ifM
is not linearly independent.
Dimension and basis
Definition 9: a vector space
X
is finite dimensional if there exists an \in \mathbb{N}
, such thatX
contains a set ofn
linearly independent vectors, while every set ofn+1
vectors inX
is linearly dependent. In this casen
is the dimension ofX
, 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 thenX
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 spaceX
is a linearly independent subset ofX
, that spansX
.
Such a set B
is also called a Hamel basis of X
.
Theorem 1: every vector space
X
has a Hamel basis.
??? note "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 subspaceM \subset X
has dimension less thann
.
??? note "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 $\dim M < n$.