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 $n$.