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Chapter 1 Linear algebra: key concepts
1.1 Vector spaces
-
Definition. A vector space \(V\) over a field \(\F \) is a set \(V\) with two operations:
- addition
-
\(V\times V\to V: (v,w)\mapsto v+w\) such that:
-
• \(v+w=w+v\), 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 \(v+(-v)=0=(-v)+v\).
In fancy language, \(V\) with addition is an abelian group.
-
- 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.