\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}

\def\Field{\mathbb{F}}

\)

#### 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.)

**Examples**

**Fun**(\(A,\Field\)), for any \(A\) – vector space over \(\Field\).…