definition of complex Euclidean vectors space
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
Target Context
- The reader will have a definition of complex Euclidean vectors space.
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:
\( d\): \(\in \mathbb{N} \setminus \{0\}\)
\(*\mathbb{C}^d\): \(= \text{ the complex Euclidean set }\) with the \(\mathbb{C}\)-scalar multiplication and the addition specified below, \(\in \{\text{ the } \mathbb{C} \text{ vectors spaces }\}\)
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Conditions:
\(\forall r = (r^1, ..., r^d) \in \mathbb{C}^d, \forall s \in \mathbb{C} (s r = (s r^1, ..., s r^d))\)
\(\land\)
\(\forall r = (r^1, ..., r^d), r' = (r'^1, ..., r'^d) \in \{\mathbb{C}^d\} (r + r' = (r^1 + r'^1, ..., r^d + r'^d))\)
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2: Note
Let us see that \(\mathbb{C}^d\) satisfies the conditions to be a \(\mathbb{C}\) vectors space.
1) \(\forall v_1, v_2 \in \mathbb{C}^d (v_1 + v_2 \in \mathbb{C}^d)\) (closed-ness under addition): \(v_1 = ({v_1}^1, ..., {v_1}^d)\) and \(v_2 = ({v_2}^1, ..., {v_2}^d)\), and \(v_1 + v_2 = ({v_1}^1 + {v_2}^1, ..., {v_1}^d + {v_2}^d) \in \mathbb{C}^d\).
2) \(\forall v_1, v_2 \in \mathbb{C}^d (v_1 + v_2 = v_2 + v_1)\) (commutativity of addition): \(v_1 = ({v_1}^1, ..., {v_1}^d)\) and \(v_2 = ({v_2}^1, ..., {v_2}^d)\), and \(v_1 + v_2 = ({v_1}^1 + {v_2}^1, ..., {v_1}^d + {v_2}^d) = ({v_2}^1 + {v_1}^1, ..., {v_2}^d + {v_1}^d) = v_2 + v_1\).
3) \(\forall v_1, v_2, v_3 \in \mathbb{C}^d ((v_1 + v_2) + v_3 = v_1 + (v_2 + v_3))\) (associativity of additions): \(v_1 = ({v_1}^1, ..., {v_1}^d)\), \(v_2 = ({v_2}^1, ..., {v_2}^d)\), and \(v_3 = ({v_3}^1, ..., {v_3}^d)\), and \((v_1 + v_2) + v_3 = ({v_1}^1 + {v_2}^1, ..., {v_1}^d + {v_2}^d) + ({v_3}^1, ..., {v_3}^d) = ({v_1}^1 + {v_2}^1 + {v_3}^1, ..., {v_1}^d + {v_2}^d + {v_3}^d) = ({v_1}^1, ..., {v_1}^d) + ({v_2}^1 + {v_3}^1, ..., {v_2}^d + {v_3}^d) = v_1 + (v_2 + v_3)\).
4) \(\exists 0 \in \mathbb{C}^d (\forall v \in \mathbb{C}^d (v + 0 = v))\) (existence of 0 element): \(0 = (0, ..., 0) \in \mathbb{C}^d\) and \(v = (v^1, ..., v^d)\), and \(v + 0 = (v^1 + 0, ..., v^d + 0) = (v^1, ..., v^d) = v\).
5) \(\forall v \in \mathbb{C}^d (\exists v' \in \mathbb{C}^d (v' + v = 0))\) (existence of inverse element): \(v = (v^1, ..., v^d)\) and \(v' := - v = (- v^1, ..., - v^d) \in \mathbb{C}^d\), and \(v' + v = (- v^1 + v^1, ..., - v^d + v^d) = (0, ..., 0) = 0\).
6) \(\forall v \in \mathbb{C}^d, \forall r \in \mathbb{C} (r . v \in \mathbb{C}^d)\) (closed-ness under scalar multiplication): \(v = (v^1, ..., v^d)\), and \(r . v = (r v^1, ..., r v^d) \in \mathbb{C}^d\).
7) \(\forall v \in \mathbb{C}^d, \forall r_1, r_2 \in \mathbb{C}^d ((r_1 + r_2) . v = r_1 . v + r_2 . v)\) (scalar multiplication distributability for scalars addition): \(v = (v^1, ..., v^d)\), and \((r_1 + r_2) . v = (r_1 + r_2) (v^1, ..., v^d) = ((r_1 + r_2) v^1, ..., (r_1 + r_2) v^d) = (r_1 v^1 + r_2 v^1, ..., r_1 v^d + r_2 v^d) = (r_1 v^1, ..., r_1 v^d) + (r_2 v^1, ..., r_2 v^d) = r_1 (v^1, ..., v^d) + r_2 (v^1, ..., v^d) = r_1 . v + r_2 . v\).
8) \(\forall v_1, v_2 \in \mathbb{C}^d, \forall r \in \mathbb{C}^d (r . (v_1 + v_2) = r . v_1 + r . v_2)\) (scalar multiplication distributability for vectors addition): \(v_1 = ({v_1}^1, ..., {v_1}^d)\) and \(v_2 = ({v_2}^1, ..., {v_2}^d)\), and \(r . (v_1 + v_2) = r ({v_1}^1 + {v_2}^1, ..., {v_1}^d + {v_2}^d) = (r ({v_1}^1 + {v_2}^1), ..., r ({v_1}^d + {v_2}^d)) = (r {v_1}^1, ..., r {v_1}^d) + (r {v_2}^1, ..., r {v_2}^d) = r ({v_1}^1, ..., {v_1}^d) + r ({v_2}^1, ..., {v_2}^d) = r . v_1 + r . v_2\).
9) \(\forall v \in \mathbb{C}^d, \forall r_1, r_2 \in \mathbb{C}^d ((r_1 r_2) . v = r_1 . (r_2 . v))\) (associativity of scalar multiplications): \(v = (v^1, ..., v^d)\), and \((r_1 r_2) . v = (r_1 r_2) . (v^1, ..., v^d) = (r_1 r_2 v^1, ..., r_1 r_2 v^d) = r_1 (r_2 v^1, ..., r_2 v^d) = r_1 . (r_2 . v)\).
10) \(\forall v \in \mathbb{C}^d (1 . v = v)\) (identity of 1 multiplication): \(v = (v^1, ..., v^d)\), and \(1 . v = 1 (v^1, ..., v^d) = (1 v^1, ..., 1 v^d) = (v^1, ..., v^d) = v\).
