Fields, vector spaces, subspaces, linear operators, range space, null space
A field \((F,+,·)\) is a set with two commutative associative operations with \(0\) and\(1\), additive and multiplicative inverses and subject to the distributive law.
Examples (Common fields)
\((\Real,+,·)\) is a field. Rational, Complex numbers. Matrices (when they are fields?)
Vector spaces – modules over a field. (Real definition: abelian group with an action of \(\Field\) and several natural axioms.)
Fun(\(A,\Field\)), for any \(A\) – vector space over \(\Field\).…