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DC Field | Value | Language |
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dc.contributor.advisor | García López, Ricardo, 1962- | - |
dc.contributor.author | Pericacho Allende, Verónica | - |
dc.date.accessioned | 2014-05-08T08:38:15Z | - |
dc.date.available | 2014-05-08T08:38:15Z | - |
dc.date.issued | 2013-07-20 | - |
dc.identifier.uri | http://hdl.handle.net/2445/53927 | - |
dc.description | Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2013, Director: Ricardo García López | ca |
dc.description.abstract | The first section of this work discusses algebras. Particularly, the algebras which interest us are division algebras, which are algebras over a field where division is always possible. Then we introduce quaternions. We show that quaternions are a non-commutative division algebra. Also we will see how can we express a quaternion through real and complex matrices of dimensions 2 and 4 respectively, and why the equation $z^2 + 1 = 0$ with $z \in \mathbb{H}$ has infinite solutions. In the last part of this section, we prove the Frobenius Theorem which affirms that the only division algebras of finite dimension over $R$ are the real numbers, the complex numbers and the quaternions. Hamilton discovered quaternions with the idea of using them to study rotations in 3-dimensional space. In the third section of this work we will see how to represent 3-dimensional rotations with unit quaternions. We will introduce the octonions in the fourth part of this work. We will see that octonions form a non-associative division algebra. In the next section we introduce the Cayley-Dickson construction for normed algebras. By this construction, we can obtain the complex numbers from the real numbers, the quaternions from the complex numbers and y the octonions from the quaternions. Finally, we will see that we can define a cross product in $\mathbb{R}^n$ only if $n$ = 1, 3 or 7. We will use this fact to prove a theorem, asserting that the possible dimensions for a normed algebra over $\mathbb{R}$ are only 1, 2, 4, 8. We will deduce from this statement a Theorem of Hurwitz which states that if $n\in\mathbb{N}$, the product of two sums of $n$ squares can be expressed as a sum of $n$ squares only if $n = 1,2, 4, 8$. | eng |
dc.format.extent | 41 p. | - |
dc.format.mimetype | application/pdf | - |
dc.language.iso | spa | ca |
dc.rights | cc-by-nc-nd (c) Verónica Pericacho Allende, 2013 | - |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/3.0/es | - |
dc.source | Treballs Finals de Grau (TFG) - Matemàtiques | - |
dc.subject.classification | Quaternions | - |
dc.subject.classification | Treballs de fi de grau | - |
dc.subject.classification | Àlgebres no associatives | ca |
dc.subject.classification | Àlgebres no commutatives | ca |
dc.subject.other | Quaternions | - |
dc.subject.other | Bachelor's theses | - |
dc.subject.other | Nonassociative algebras | eng |
dc.subject.other | Noncommutative algebras | eng |
dc.title | Cuaterniones y octoniones | ca |
dc.type | info:eu-repo/semantics/bachelorThesis | ca |
dc.rights.accessRights | info:eu-repo/semantics/openAccess | ca |
Appears in Collections: | Treballs Finals de Grau (TFG) - Matemàtiques |
Files in This Item:
File | Description | Size | Format | |
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memoria.pdf | Memòria | 442.46 kB | Adobe PDF | View/Open |
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