book:chap2:2.7_the_inner_product
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| book:chap2:2.7_the_inner_product [2022/04/01 21:01] – jv | book:chap2:2.7_the_inner_product [2024/12/03 23:28] (current) – jv | ||
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| b) linearity in the first argument: | b) linearity in the first argument: | ||
| - | $\langle c \, \mathbf v \mid \mathbf w \rangle = c \; \langle \mathbf v \mid \mathbf w \rangle$ \\ | + | $\langle c \odot \mathbf v \mid \mathbf w \rangle = c \; \langle \mathbf v \mid \mathbf w \rangle$ \\ |
| - | and $\langle \mathbf u + \mathbf v \mid \mathbf w \rangle = \langle \mathbf u \mid \mathbf w \rangle + \langle \mathbf v \mid \mathbf w \rangle$\\ | + | and $\langle \mathbf u \oplus |
| c) positivity: | c) positivity: | ||
| Line 41: | Line 41: | ||
| $\langle \mathbf v \mid \mathbf v \rangle = 0 \; | $\langle \mathbf v \mid \mathbf v \rangle = 0 \; | ||
| - | For a vector space over $\mathbb{C}$ the requirement a) is replaced by | + | For a vector space over $\mathbb{C}$ |
| + | Moreover, | ||
| a) conjugate symmetry: | a) conjugate symmetry: | ||
| $\langle \mathbf v \mid \mathbf w \rangle = \overline{\langle \mathbf w \mid \mathbf v \rangle}$ | $\langle \mathbf v \mid \mathbf w \rangle = \overline{\langle \mathbf w \mid \mathbf v \rangle}$ | ||
| - | and the constant | + | such that $\langle \mathbf v \mid \mathbf v \rangle$ is a real number. |
| </ | </ | ||
| Line 58: | Line 59: | ||
| Conjugate symmetry and linearity for the first argument imply the following relations for the second argument | Conjugate symmetry and linearity for the first argument imply the following relations for the second argument | ||
| \begin{align*} | \begin{align*} | ||
| - | \langle \mathbf v \mid c \, \mathbf w \rangle | + | \langle \mathbf v \mid c \odot \mathbf w \rangle |
| - | &= \overline{\langle c \, \mathbf w \mid \mathbf v \rangle} | + | &= \overline{\langle c \odot \mathbf w \mid \mathbf v \rangle} |
| = \bar{c} \; | = \bar{c} \; | ||
| = \bar{c} \; \langle \mathbf v \mid \mathbf w \rangle | = \bar{c} \; \langle \mathbf v \mid \mathbf w \rangle | ||
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| - | \langle \mathbf u \mid \mathbf v + \, \mathbf w \rangle | + | \langle \mathbf u \mid \mathbf v \oplus \mathbf w \rangle |
| - | &= \overline{\langle \mathbf v + \mathbf w \mid \mathbf u \rangle} | + | &= \overline{\langle \mathbf v \oplus |
| = \overline{\langle \mathbf v \mid \mathbf u \rangle} + \overline{\langle \mathbf w \mid \mathbf u \rangle} | = \overline{\langle \mathbf v \mid \mathbf u \rangle} + \overline{\langle \mathbf w \mid \mathbf u \rangle} | ||
| = \langle \mathbf u \mid \mathbf v \rangle + \langle \mathbf u \mid \mathbf w \rangle | = \langle \mathbf u \mid \mathbf v \rangle + \langle \mathbf u \mid \mathbf w \rangle | ||
| \end{align*} | \end{align*} | ||
| - | |||
| </ | </ | ||
| Line 110: | Line 110: | ||
| c^2 = a^2 + b^2 - 2\, | c^2 = a^2 + b^2 - 2\, | ||
| \end{align*} | \end{align*} | ||
| - | Let now $a$, $b$, and $c$ be the length of the vectors $\mathbf a$, $\mathbf b$ and $\mathbf c = \mathbf | + | Let now $a$, $b$, and $c$ be the length of the vectors $\mathbf a$, $\mathbf b$ and $\mathbf c = \mathbf |
| as shown in [[# | as shown in [[# | ||
| Then we have | Then we have | ||
| Line 116: | Line 116: | ||
| a^2 + b^2 - 2\, | a^2 + b^2 - 2\, | ||
| &= c^2 = \mathbf c \cdot \mathbf c | &= c^2 = \mathbf c \cdot \mathbf c | ||
| + | = (\mathbf b - \mathbf a) \cdot (\mathbf b - \mathbf a) | ||
| = (\mathbf a - \mathbf b) \cdot (\mathbf a - \mathbf b) | = (\mathbf a - \mathbf b) \cdot (\mathbf a - \mathbf b) | ||
| \\ | \\ | ||
| &= \mathbf a \cdot \mathbf a - 2\, \mathbf a \cdot \mathbf b + \mathbf b \cdot \mathbf b | &= \mathbf a \cdot \mathbf a - 2\, \mathbf a \cdot \mathbf b + \mathbf b \cdot \mathbf b | ||
| = a^2 + b^2 - 2\, \mathbf a \cdot \mathbf b | = a^2 + b^2 - 2\, \mathbf a \cdot \mathbf b | ||
| - | \\ | + | \\[2mm] |
| \Rightarrow\quad | \Rightarrow\quad | ||
| - | \mathbf a \cdot \mathbf b = |\mathbf a| \, |\mathbf b| \, \cos\theta | + | \mathbf a \cdot \mathbf b &= |\mathbf a| \, |\mathbf b| \, \cos\theta |
| \\ | \\ | ||
| \end{align*} | \end{align*} | ||
book/chap2/2.7_the_inner_product.1648839686.txt.gz · Last modified: 2022/04/01 21:01 by jv