Productum interius

Productum interius duorum vectorum

Productum interius seu productum scalare seu puncti productum est productum duorum vectorum ${\vec {a}}$ et ${\vec {b}}$ ubi singulus numerus scalaris producitur, quid datur formula

${\vec {a}}\cdot {\vec {b}}=\left\|{\vec {a}}\right\|\,\left\|{\vec {b}}\right\|\cos \theta \,$

Quod productum valorem zerum attingit cum duo vectores perpendiculares sunt et maximum, cum duo vectores paralleli sunt, aequantem magnitudines duorum vectorum multiplicatos.

Coordinatis orthogonalibus et valoribus realibus

His vectoribus iuxta basem orthogonalem scriptis

${\vec {a}}={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\\vdots \end{bmatrix}}\,{\textit {et}}\;\;{\vec {b}}={\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\\\vdots \end{bmatrix}}$,

productum scribi potest

$\langle {\vec {a}},{\vec {b}}\rangle ={\vec {a}}^{T}\,{\vec {b}}={\begin{bmatrix}a_{1}\,a_{2}\,a_{3}\,\dots \end{bmatrix}}{\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\\\vdots \end{bmatrix}}=\sum _{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$

ubi T denotat transpositionem matricis, Σ denotat summam arithmeticam et n est dimensio spatii vectorialis.

Coordinatis orthogonalibus et valoribus complexis

His autem vectoribus valoribus complexis praeditis, productum interius scribi oportet

$\langle {\vec {a}},{\vec {b}}\rangle ={\vec {a}}^{\dagger }\,{\vec {b}}={\begin{bmatrix}a_{1}^{*}\,a_{2}^{*}\,a_{3}^{*}\,\dots \end{bmatrix}}{\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\\\vdots \end{bmatrix}}=\sum _{i=1}^{n}a_{i}^{*}b_{i}=a_{1}^{*}b_{1}+a_{2}^{*}b_{2}+\cdots +a_{n}^{*}b_{n}$

ubi * denotat coniugationem complexam et † denotat simultaneam coniugationem et transpositionem. Hac definitione maxime numeris complexis accomodata effecit ut semper scribi possit valore scalari reali

${\vec {a}}\cdot {\vec {a}}=\left\|{\vec {a}}\right\|^{2}$

Bibliographia

• Anton, Howard. 1977. Elementary Linear Algebra. Novi Eboraci: John Wiley & Sons.
• Birkhoff, Garrett, et Saunders MacLane. 1965. A Survey of Modern Algebra. Editio tertia. Novi Eboraci: Macmillan.
• Bourbaki, Nicolas Bourbaki. 2007. Algèbre, chapitres 1 à 3 Éléments de mathematique. Berolini: Springer Verlag.
• Gowers, Timothy, ed. 2008. The Princeton Companion to Mathematics. Princeton: Princeton University Press. ISBN 978-0-691-11880-2.
• Hart, Roger. 2011. The Chinese Roots of Linear Algebra. Baltimore: Johns Hopkins University Press. ISBN 978-0-8018-9755-9.
• Heffron, Jim. 2011. Linear Algebra. Liber ab auctore editus, in rete