Vector Space

algebra linear-algebra

Definition

Vector Space

Let $F$ be a field. A vector space over $F$ is an Abelian group $(V,+)$ equipped with a scalar multiplication $\cdot :F\times V\to V$.

Elements of $V$ are called vectors; elements of $F$ are called scalars.

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 (Vector Space)

Let $V$ be a vector space. A set $B\subseteq V$ is a basis of $V$ if $B$ is linearly independent and $[B]=V$, where $[B]$ denotes the linear span of $B$.

Every $v\in V$ can be uniquely expressed as a linear combination of vectors from $B$. The coefficients of this combination are the coordinates of $v$ with respect to $B$.

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Basis Selection Lemma

Definition

Basis Selection Lemma

Let $B=\{v_1,\dots ,v_n\}$ be a basis of a vector space $V$ and let

$$a=\sum _{i=1}^n{\mu }_i{v}_i$$

be a linear combination with ${\mu }_j\ne 0$ for some $j$. Then the exchanged set

$$B^{\prime }=(B\setminus \{v_j\})\cup \{a\}$$

is also a basis of $V$. In particular, $B^{\prime }$ is linearly independent and $[B^{\prime }]=[B]$, where $[\cdot ]$ denotes the linear span.

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Canonical Basis

Definition

Canonical Basis

Let $V$ be a $n$-dimensional vector space. The canonical basis $\mathcal{B}$ defined as:

$$\mathcal{B}=\{e_1,e_2,\dots ,e_n\}$$

where each vector $e_j$ has a $1$ in the $j$-th position and $0$ elsewhere:

$$e_1=\left[\begin{array}{c}1\\ 0\\ ⋮\\ 0\end{array}\right],e_2=\left[\begin{array}{c}0\\ 1\\ ⋮\\ 0\end{array}\right],\dots ,e_n=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 1\end{array}\right]$$

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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 basis, 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:

$$V\cong F^n$$