definition of complex conjugate of complex 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 conjugate of complex 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:
\( \mathbb{C}\): with the canonical field structure
\( V\): \(\in \{\text{ the } \mathbb{C} \text{ vectors spaces }\}\), with any addition, \(+\), and any scalar multiplication, \(*\)
\( \overline{V}\): \(= V\), \(\in \{\text{ the } \mathbb{C} \text{ vectors spaces }\}\), with the addition, \(\oplus\), and the scalar multiplication, \(\otimes\), specified below
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Conditions:
\(\forall v, v' \in \overline{V} (v \oplus v' = v + v')\)
\(\land\)
\(\forall v \in \overline{V}, \forall c \in \mathbb{C} (c \otimes v = \overline{c} * v)\)
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2: Note
\(\overline{V} = V\) means that they are the same sets-wise.
There is the bijective correspondence, \(f: V \to \overline{V}, v \mapsto v\), which is not any vectors space homomorphism: \(f (c * v) = c * v = \overline{c} \otimes v = \overline{c} \otimes f (v)\), which is not linear.
\(f (v)\) is called "conjugate of \(v\)".
When \(V\) is \(d\)-dimensional and has a basis, \(B = \{b_1, ..., b_d\}\), \(B\) is also a basis for \(\overline{V}\): for \(c^1 \otimes b_1 \oplus ... \oplus c^d \otimes b_d = 0\), \(\overline{c^1} * b_1 + ... + \overline{c^d} * b_d = 0\), which implies that \(\overline{c^1} = ... = \overline{c^d} = 0\), which implies that \(c^1 = ... = c^d = 0\), so, \(B\) is linearly independent on \(\overline{V}\); for each \(v \in \overline{V}\), \(v = v^1 * b_1 + ... + v^d * b_d = \overline{v^1} \otimes b_1 \oplus ... \oplus \overline{v^d} \otimes b_d\), so, \(B\) spans \(\overline{V}\).
The components set of any \(v \in V\) with respect to \(B\) is \((v^1, ..., v^d)^t\), which means that \(v = v^1 * b_1 + ... + v^d * b_d\), which means that \(v = \overline{v^1} \otimes b_1 \oplus ... \oplus \overline{v^d} \otimes b_d\), so, the components set of \(v \in \overline{V}\) with respect to \(B\) is \((\overline{v^1}, ..., \overline{v^d})\), which is frequently denoted as the row vector, as a custom.
When \(V\) has an inner product, \(\langle \bullet, \bullet \rangle_V\), \(\langle \bullet, \bullet \rangle: \overline{V} \times V \to \mathbb{C}, (v', v) \mapsto \langle v, f^{-1} (v') \rangle_V\) is often called "inner product", although the definition of inner product on real or complex vectors space is on a single vectors space.
For \(v' = v'^j \otimes b_j\) and \(v = v^l * b_l\), \(\langle v', v \rangle = \langle v'^j \otimes b_j, v^l * b_l \rangle = \langle v^l * b_l, f^{-1} (v'^j \otimes b_j) \rangle_V = \langle v^l * b_l, \overline{v'^j} * b_j \rangle_V = \overline{v'^j} v^l \langle b_l, b_j \rangle_V\), and especially if the basis is orthonormal, \(= \overline{v'^j} v^l \delta_{l, j} = \sum_j \overline{v'^j} v^j = (\overline{v'^1}, ..., \overline{v'^d}) (v^1, ..., v^d)^t\).
Let us see that \(\overline{V}\) is indeed a \(\mathbb{C}\) vectors space.
1) for any elements, \(v_1, v_2 \in \overline{V}\), \(v_1 \oplus v_2 \in \overline{V}\) (closed-ness under addition): \(v_1 \oplus v_2 = v_1 + v_2 \in V = \overline{V}\).
2) for any elements, \(v_1, v_2 \in \overline{V}\), \(v_1 \oplus v_2 = v_2 \oplus v_1\) (commutativity of addition): \(v_1 \oplus v_2 = v_1 + v_2 = v_2 + v_1 = v_2 \oplus v_1\).
3) for any elements, \(v_1, v_2, v_3 \in \overline{V}\), \((v_1 \oplus v_2) \oplus v_3 = v_1 \oplus (v_2 \oplus v_3)\) (associativity of additions): \((v_1 \oplus v_2) \oplus v_3 = (v_1 + v_2) + v_3 = v_1 + (v_2 + v_3) = v_1 \oplus (v_2 \oplus v_3)\).
4) there is a 0 element, \(0 \in \overline{V}\), such that for any \(v \in \overline{V}\), \(v \oplus 0 = v\) (existence of 0 vector): \(0 \in V = \overline{V}\), and \(v \oplus 0 = v + 0 = v\).
5) for any element, \(v \in \overline{V}\), there is an inverse element, \(v' \in \overline{V}\), such that \(v' \oplus v = 0\) (existence of inverse vector): \(v' := -v \in V = \overline{V}\), and \(v' \oplus v = - v + v = 0\).
6) for any element, \(v \in \overline{V}\), and any scalar, \(r \in \mathbb{C}\), \(r \otimes v \in \overline{V}\) (closed-ness under scalar multiplication): \(r \otimes v = \overline{r} * v \in V = \overline{V}\).
7) for any element, \(v \in \overline{V}\), and any scalars, \(r_1, r_2 \in \mathbb{C}\), \((r_1 + r_2) \otimes v = r_1 \otimes v \oplus r_2 \otimes v\) (scalar multiplication distributability for scalars addition): \((r_1 + r_2) \otimes v = \overline{(r_1 + r_2)} * v = (\overline{r_1} + \overline{r_2}) * v = \overline{r_1} * v + \overline{r_2} * v = r_1 \otimes v \oplus r_2 \otimes v\).
8) for any elements, \(v_1, v_2 \in \overline{V}\), and any scalar, \(r \in \mathbb{C}\), \(r \otimes (v_1 \oplus v_2) = r \otimes v_1 \oplus r \otimes v_2\) (scalar multiplication distributability for vectors addition): \(r \otimes (v_1 \oplus v_2) = \overline{r} * (v_1 + v_2) = \overline{r} * v_1 + \overline{r} * v_2 = r \otimes v_1 \oplus r \otimes v_2\).
9) for any element, \(v \in \overline{V}\), and any scalars, \(r_1, r_2 \in \mathbb{C}\), \((r_1 r_2) \otimes v = r_1 \otimes (r_2 \otimes v)\) (associativity of scalar multiplications): \((r_1 r_2) \otimes v = \overline{(r_1 r_2)} * v = (\overline{r_1} \overline{r_2}) * v = \overline{r_1} * (\overline{r_2} * v) = r_1 \otimes (r_2 \otimes v)\).
10) for any element, \(v \in \overline{V}\), \(1 \otimes v = v\) (identity of 1 multiplication): \(1 \otimes v = \overline{1} * v = 1 * v = v\).
