Vector Spaces

Generally

Section 30 Vector Spaces

30.1 Definition: Vector Space

$⟨V,+,\cdot ⟩$ is a vector space over $F$ if: for all $a,b\in F$ and $\alpha ,\beta \in V$

• $+:V\times V\to V$ a function (addition)
• $\cdot :\mathbb{R}\times V\to V$ a function (scalar multiplication) Example: $⟨{\mathbb{R}}^n,+,\cdot ⟩$
• ${\mathbb{R}}^n=\underset{n}{\underbrace{\mathbb{R}\times \mathbb{R}\times \cdots \times \mathbb{R}}}$

30.6 Definition: Span

If all vectors in the vector space can be expressed as a linear combination of a set of vectors in the vector space, this vector set span (or generate ) this vector space. If this is a minimal subset, it is a basis.

Basis

$B$ is a basis if $B$ is linearly independent and $\mathbf{Span}(B)=V$

30.9 Definition: finite dimensional

A First Course in Abstract Algebra, p.277

A vector space $V$ over a field $F$ is finite dimensional if there is a finite subset of $V$ whose vectors span $V$.

30.11 Example

A First Course in Abstract Algebra, p.277

If $F\le E$ and $\alpha \in E$ is algebraic over the field $F$, then $F(\alpha )$ is a finite-dimensional vector space over $F$.

By 29.18 Theorem $F(\alpha )$ is spanned by the vectors in $\{1,\alpha ,\cdots ,{\alpha }^{n-1}\}$, where $n=deg(\alpha ,F)$. See $Q(\sqrt{2})$ in 29.16 Example

If $n=deg(\alpha ,F)$, then every vector can be uniquely expressed by $\{1,\alpha ,\cdots ,{\alpha }^{n-1}\}$

30.21 Definition: Dimension

If $V$ is a finite-dimensional vector space over a field $F$, the number of elements in a basis is the dimension of $V$ over $F$.

30.22 Example

A First Course in Abstract Algebra, p.280

Let $E$ be an extension field of a field $F$, and let $\alpha \in E$. If $\alpha$ is algebraic over $F$ and $deg(\alpha ,F)=n$, then the dimension of $F(\alpha )$ as a vector space over $F$ is $n$.

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In Linear Algebra

The eight axioms

Some ‘obvious’ facts

• There is only one zero vector. Proof: Take two zero vectors $0$ and $0^{\prime }$ $0^{\prime }=0^{\prime }+0$ (3. axiom) $=0+0^{\prime }$ (1. axiom) $=0$ (3. axiom)
• Each $\mathbf{v}$ has only one negative vector

Subspaces

Let $V$ be a vector space. A nonempty subset $U\subseteq V$ is called a subspace of $V$ if

• $\mathbf{v}+\mathbf{w}\in U$
• $\lambda \mathbf{v}\in U$ and $U$ is also a vector space

Examples

$U$ is all quadratic polynomials $\mathbf{p}=p_0+p_{1}x+p_2{x}^2$ This subspace is isomorphic to ${\mathbb{R}}^3$ Prove isomorphism

NOTE

a polynomial should have finite terms

• $U$ all matrices of trace $0$ (What is trace of a matrix)

Lemma 4.16

Ever linear combination of $V$ is again in $V$

Title

It has to be finite linear combinations

Examples of Basis

Vector sapce $V$	basis $B$
${\mathbb{R}}^m$	$\{e_1,e_2,\dots ,e_m\}$
$\mathbf{C}(A)$	independent columns of $A$
$2\times 2$ symmetric matrices (Subspace of $R^{2\times 2}$)	$\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\}$
$\mathbb{R}[x]$ (polynomials)	$\{x^i:i=0,1,\dots \}$ (infinite set)
$0$ (smallest vector space)	$\varnothing$ (emptyset)

Definition: finitely generated vector space

A vector space $V$ is called finitely generated if there exists a finite subset $G\subseteq V$ with $\text{Span}(G)=V$

${\mathbb{R}}^m$ is finitely generated $\mathbb{R}[x]$ is not finitely generated

Theorem 4.22

let $G\subseteq V$ be a finite subset with $\text{Span}(G)=V$. Then $V$ has a basis $B\subseteq G$

Steinitz exchange lemma

(ii) means: one can enlarge $F$ by some elements from $G$ such that the result has at most the size of $G$ and also spans $V$

All bases have the same size

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Definition: Dimension

Definition

$\text{dim}(V)$ is the size of an arbitrary basis $B$ of $V$

Linear transformations between vector spaces

NOTE

If a linear transformation $T:V\to W$ is bijective. Then $V$ and $W$ are isomorphic (Prove isomorphism)

Here, isomorphic means the bases and dimension are preserved See Lemma 4.27

Example

$V={\mathbb{R}}^{2\times 2}$, $W={\mathbb{R}}^4$, $T:\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\to \left(\begin{array}{c}a\\ b\\ c\\ d\end{array}\right)$ or generally $V={\mathbb{R}}^{m\times n}$, $W={\mathbb{R}}^{mn}$ This is bijective (invertible)

$V=$ polynomials of degree 2, $W={\mathbb{R}}^3$, $T:p_0+p_{1}x+p_2{x}^2\to \left(\begin{array}{c}{p}_0\\ p_1\\ p_3\end{array}\right)$

NOTE

All m-dimensional (real) vector spaces are isomorphic

How to compute the subspaces

Computing the three fundamental subspaces