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45 lines
1.4 KiB
Markdown
45 lines
1.4 KiB
Markdown
[[operazione]] tra [[array|vettori]].
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$$
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\Huge
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\braket{\mathbf{x}, \mathbf{y}}
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\qquad
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\mathbf{x} \cdot \mathbf{y}
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$$
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Consiste nel [[prodotto|moltiplicare]] i componenti uno ad uno tra loro, e poi [[somma|sommare]] tutti i risultati:
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$$
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\Large
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\def \varW {{\color{Yellow} w}}
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\def \varX {{\color{Red} \mathbf{x}}}
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\def \varY {{\color{MediumBlue} \mathbf{y}}}
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\def \varZ {{\color{Lime} \mathbf{z}}}
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\def \varXa {{\color{Crimson} x_1}}
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\def \varXb {{\color{IndianRed} x_2}}
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\def \varXc {{\color{Salmon} x_3}}
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\def \varXn {{\color{LightSalmon} x_n}}
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\def \varYa {{\color{RoyalBlue} y_1}}
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\def \varYb {{\color{DeepSkyBlue} y_2}}
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\def \varYc {{\color{LightSkyBlue} y_3}}
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\def \varYn {{\color{PaleTurquoise} y_n}}
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\braket{\varX, \varY}
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=
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(\varXa \cdot \varYa) +
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(\varXb \cdot \varYb) +
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(\varXc \cdot \varYc) +
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\dots +
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(\varXn \cdot \varYn)
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$$
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## Proprietà in $\mathbb{R}^n$
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- $\braket{\varX, \varY} = \braket{\varY, \varX}$ ([[proprietà commutativa]])
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- $\braket{\varX, (\varY + \varZ)} = \braket{\varX, \varY} + \braket{\varX, \varZ}$ ([[proprietà distributiva]])
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- $\varW \cdot \braket{\varX, \varY} = \braket{(\varW \cdot \varX), \varY} = \braket{\varX, (\varW \cdot \varY)}$
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- $| \varX | + | \varY | \geq | \braket{\varX, \varY} |$ ([[disuguaglianza di Cauchy-Schwarz]])
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- $\braket{\varX, \varX} = | \varX |^2$
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## Proprietà in $\mathbb{R}^2$
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- $\braket{\varX, \varY} = |\varX| \cdot |\varY| \cdot \cos(angolo\ tra\ \varX\ e\ \varY)$
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