description/proof of that complex vectors space can be regarded to be canonical real 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 description and a proof of the proposition that any complex vectors space can be regarded to be the canonical real 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:
\(V\): \(\in \{\text{ the complex vectors spaces }\}\) with any scalar multiplication, \(.: \mathbb{C} \times V \to V\), and any addition, \(+: V \times V \to V\)
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Statements:
\(V \in \{\text{ the real vectors spaces }\}\) with the restricted scalar multiplication, \(.': \mathbb{R} \times V \to V = . \vert_{\mathbb{R} \times V}\), and the addition, \(+: V \times V \to V\)
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2: Proof
Whole Strategy: Step 1: see that \(V\) with \(.'\) and \(+\) satisfies the conditions to be a vectors space.
Step 1:
Let us see that \(V\) with \(.'\) and \(+\) satisfies the conditions to be a vectors space.
1) for any elements, \(v_1, v_2 \in V\), \(v_1 + v_2 \in V\) (closed-ness under addition): \(+\) for the real vectors space is the same with \(+\) for the complex vectors space, while \(v_1 + v_2 \in V\) is satisfied for the complex vectors space.
2) for any elements, \(v_1, v_2 \in V\), \(v_1 + v_2 = v_2 + v_1\) (commutativity of addition): \(+\) for the real vectors space is the same with \(+\) for the complex vectors space, while \(v_1 + v_2 = v_2 + v_1\) is satisfied for the complex vectors space.
3) for any elements, \(v_1, v_2, v_3 \in V\), \((v_1 + v_2) + v_3 = v_1 + (v_2 + v_3)\) (associativity of additions): \(+\) for the real vectors space is the same with \(+\) for the complex vectors space, while \((v_1 + v_2) + v_3 = v_1 + (v_2 + v_3)\) is satisfied for the complex vectors space.
4) there is a 0 element, \(0 \in V\), such that for any \(v \in V\), \(v + 0 = v\) (existence of 0 vector): \(+\) for the real vectors space is the same with \(+\) for the complex vectors space, while \(v + 0 = v\) is satisfied for the complex vectors space.
5) for any element, \(v \in V\), there is an inverse element, \(v' \in V\), such that \(v' + v = 0\) (existence of inverse vector): \(+\) for the real vectors space is the same with \(+\) for the complex vectors space, while \(v' + v = 0\) is satisfied for the complex vectors space.
6) for any element, \(v \in V\), and any scalar, \(r \in \mathbb{R}\), \(r .' v \in V\) (closed-ness under scalar multiplication): \(r \in \mathbb{C}\), and \(r .' v = r . v \in V\).
7) for any element, \(v \in V\), and any scalars, \(r_1, r_2 \in \mathbb{R}\), \((r_1 + r_2) . v = r_1 . v + r_2 . v\) (scalar multiplication distributability for scalars addition): \(r_1, r_2 \in \mathbb{C}\), \(r_1 + r_2 \in \mathbb{C}\), and \((r_1 + r_2) .' v = (r_1 + r_2) . v = r_1 . v + r_2 . v = r_1 .' v + r_2 .' v\).
8) for any elements, \(v_1, v_2 \in V\), and any scalar, \(r \in \mathbb{R}\), \(r . (v_1 + v_2) = r . v_1 + r . v_2\) (scalar multiplication distributability for vectors addition): \(r \in \mathbb{C}\), and \(r .' (v_1 + v_2) = r . (v_1 + v_2) = r . v_1 + r . v_2 = r .' v_1 + r .' v_2\).
9) for any element, \(v \in V\), and any scalars, \(r_1, r_2 \in \mathbb{R}\), \((r_1 r_2) . v = r_1 . (r_2 . v)\) (associativity of scalar multiplications): \(r_1, r_2 \in \mathbb{C}\), \(r_1 r_2 \in \mathbb{C}\), and \((r_1 r_2) .' v = (r_1 r_2) . v = r_1 . (r_2 . v) = r_1 .' (r_2 .' v)\).
10) for any element, \(v \in V\), \(1 . v = v\) (identity of 1 multiplication): \(1 \in \mathbb{C}\), and \(1 .' v = 1 . v = v\).
