Chapter 1 Linear algebra: concepts and examples

1.1 Vector spaces

  • Definition. A vector space \(V\) over a field \(\F \) is a set \(V\) with two operations:

    addition

    \(V\times V\to V: (v,w)\mapsto v+w\) with respect to which \(V\) is an abelian group:

    • • \(v+w=w+v\), for all \(v,w\in V\);

    • • \(u+(v+w)=(u+v)+w\), for all \(u,v,w\in V\);

    • • there is a zero element \(0\in V\) for which \(v+0=v=0+v\), for all \(v\in V\);

    • • each element \(v\in V\) has an additive inverse \(-v\in V\) for which \(v+(-v)=0=(-v)+v\).

    scalar multiplication

    \(\F \times V\to V: (\lambda ,v)\mapsto \lambda v\) such that

    • • \((\lambda +\mu )v=\lambda v+\mu v\), for all \(v\in V\), \(\lambda ,\mu \in \F \).

    • • \(\lambda (v+w)=\lambda v+\lambda w\), for all \(v,w\in V\), \(\lambda \in \F \).

    • • \((\lambda \mu )v=\lambda (\mu v)\), for all \(v\in V\), \(\lambda ,\mu \in \F \).

    • • \(1v=v\), for all \(v\in V\).

    We call the elements of \(\F \) scalars and those of \(V\) vectors.