definition of linear map
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of map.
Target Context
- The reader will have a definition of linear map.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( V_1\): \(\in \{\text{ the vectors spaces over } F\}\)
\( V_2\): \(\in \{\text{ the vectors spaces over } F\}\)
\(*f\): \(: V_1 \to V_2\)
//
Conditions:
\(\forall r_1, r_2 \in F, \forall v_1, v_2 \in V_1 (f (r_1 v_1 + r_2 v_2) = r_1 f (v_1) + r_2 f (v_2))\).
//
2: Natural Language Description
For any field, \(F\), and any vectors spaces, \(V_1, V_2\), over \(F\), any map, \(f: V_1 \to V_2\), such that for any \(r_1, r_2 \in F\) and any \(v_1, v_2 \in V_1\), \(f (r_1 v_1 + r_2 v_2) = r_1 f (v_1) + r_2 f (v_2)\)
3: Note
The fields of \(V_1\) and \(V_2\) have to be the same in order for \(r_1 f (v_1) + r_2 f (v_2)\) to make sense; there may be an argument that the field of \(V_2\) could be a superset of that of \(V_1\), but we do not see any immediate necessity to include that case. For example, the complex numbers field is not exactly any superset of the real numbers field (\(1 + 0 i \in \mathbb{C}\) and \(1 \in \mathbb{R}\) are not exactly same according to the standard definitions: intuitively speaking, \((1, 0)\) and \(1\) are not same), although there is the canonical map from \(\mathbb{R}\) into \(\mathbb{C}\).
The \(0\) vector is inevitably mapped to the \(0\) vector, because \(f (0) = f (0 v) = 0 f (v) = 0\) for any \(v \in V_1\).
Any linear map is nothing but a 'vectors spaces' homomorphism.
