Vector Space

algebra linear-algebra

Definition

Vector Space

A vector space $(V,+,F)$, or linear space, is an algebraic structure over a field $(F,+,\cdot )$, called the scalar field, and an Abelian group $(V,+)$. A vector space that has the following properties:

Let $\lambda ,\mu \in F$ and $\mathbf{x},\mathbf{y}\in V$:

1. $\lambda \cdot (\mathbf{x}+\mathbf{y})=\lambda \cdot \mathbf{x}+\lambda \cdot \mathbf{y}$
2. $(\lambda +\mu )\cdot \mathbf{x}=\lambda \cdot \mathbf{x}+\mu \cdot \mathbf{x}$
3. $(\lambda \cdot \mu )\cdot \mathbf{x}=\lambda \cdot (\mu \cdot \mathbf{x})$
4. $1\cdot \mathbf{x}=\mathbf{x}$

Elements of a vector space $(V,+,F)$ are called vectors.

Subspace

Unterraum oder Teilraum

Subspace

A vector space $(W,+,F)$ is called subspace of vector space $(V,+,F)$, denoted as $W\le V$, if:

$$W\le V:⟺W\subseteq V\wedge W\ne \varnothing$$

The following is called subspace generated by vector $v\in V$, where $v\ne v_0$:

$$[v]=\{\lambda v\mid \lambda \in F\}$$

Cospace

Nebenraum

Cospace

Let $U$ be a subspace of a vector space $V$ and ${\mathbf{x}}_0\in V$.

$$N={\mathbf{x}}_0+U=\{{\mathbf{x}}_0+\mathbf{u}\mid \mathbf{u}\in U\}$$

Where $N$ is called cospace of $U$ at ${\mathbf{x}}_0$.

Basis

Definition

Basis

A subset $B$ of a vector space $V$ is called basis of $V$ if $B$ is linearly independent and the linear span $[B]=V$.

Each evector $v\in V$ can be uniquely expresses as a linear combination of the basis vectors. The coefficients of this linear combination are called coordinates of $v$ w.r.t. basis $B$.

The number of basis vectors |B| is called the dimension of $V$ and is dentoed as:

$$\dim V=|B|$$

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Basis Selection Lemma

Definition

Basis Selection Lemma

Let $M=\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}$ be the basis of a vector space.

$$\mathbf{a}={\mu }_1\cdot {\mathbf{v}}_1+\cdots +{\mu }_n\cdot {\mathbf{v}}_n$$

a linear combination of $M$.

Further, let $M^{\prime }$ be:

$$\begin{array}{rl}{M}^{\prime }& =(M\setminus \{{\mathbf{v}}_j\})\cup \{\mathbf{a}\}\\ & =\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_{j-1},\mathbf{a},{\mathbf{v}}_{j+1},\dots ,{\mathbf{v}}_n\}\end{array}$$

for $j$ with ${\mu }_j\ne 0$.

Then $M$ is linearly independent if and only if $M^{\prime }$ is linearly independent, and it always holds that:

$$[M^{\prime }]=[M]$$

where $[\cdot ]$ denotes the linear span.

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Canonical Basis

Definition

Coordinate Projection

A projection ${\Phi }_B:V\to F^n$ maps a vector $v\in V$ to its coordinates w.r.t. to the basis $B$:

$${\Phi }_B(\mathbf{v})=\left[\begin{array}{c}{\lambda }_1\\ {\lambda }_2\\ ⋮\\ {\lambda }_n\end{array}\right]$$

Due to the uniqueness of the representation as a linear combination of a bvasis, the coordinate ${\Phi }_B$ is bijective and satisfies the isomorphic properties:

$$\begin{array}{rl}{\Phi }_B(\mathbf{x}+\mathbf{y})& ={\Phi }_B(\mathbf{x})+{\Phi }_B(\mathbf{y})\\ {\Phi }_B(\lambda \cdot \mathbf{x})& =\lambda \cdot {\Phi }_B(\mathbf{x})\end{array}$$

with $\mathbf{x},\mathbf{y}\in V$ and $\lambda \in F$.

Thus, all arithmetic operations on $V$ can be performed in vector space $F^n$. $V$ and $F^n$ have the same structure, hence, they are called isomorphic:

$$V\cong F^n$$