| Sažetak | U ovom smo radu proučavali neke od varijanti Pitagorinog teorema. Nakon povijesti Pitagore, dokazali smo Pitagorin teorem na četiri različita načina. Prvi dokaz je bio preko sličnosti trokuta. Zatim su slijedili Euklidov dokaz, Choupeijev te Garfildov dokaz. Predstavili smo neke primjene Pitagorinog teorema u matematici, zatim i u fizici, računajući rezultantu. Naša prva varijanta Pitagorinog teorema, pokazuje da ako zamijenimo stranice trokuta vektorima u smjeru ortogonalnih osi, te hipotenuzu vektorom \(x\) Pitagorin teorem vrijedi. Druga varijanta, iskazuje da projekcija vektora \(x\) na osi određene jediničnim vektorima, omogućuje da vektor \(x\) izrazimo kao linearnu kombinaciju jediničnih vektora. Treća varijanta, proširuje prethodnu tvrdnju na konačno dimenzionalan unitaran prostor bilo koje dimenzije. Tesarski teorem, tj. obrat Pitagorinog teorema je naša četvrta varijanta. Peta varijanta je još jedno poopćenje Pitagorinog teorema iskazanog koristeći projekciju vektora na ortogonalne osi. Šesta i sedma varijanta (propozicija 2.2.2 i propozicija 2.2.3) daje nam odgovor na pitanje, može li se nešto od prethodnog primijeniti na ortonormirane baze u višedimenzionalnim prostorima. Dok je osma varijanta (propozicija 2.2.4) malo preformulirana sedma varijanta. Obrat prethodne varijante je naša deveta varijanta. Deseta varijanta nam daje odgovor na pitanje postoji li m-dimenzionalan potprostor od \(U\) na kojem projekcije elemenata baze imaju duljinu \(m\). U zadnjem poglavlju, fokusirali smo se na neke varijante Pitagorinog teorema s operatorima i matricama. Tako, naša jedanaesta varijanta pokazuje da je trag projekcije m-dimenzionalnog potprostora jednaka \(m\). Dvanaesta varijanta pokazuje da je uređena n-torka \( \langle a_1, a_2, \dots a_n \rangle \) brojeva iz intervala [0, 1], čija je suma jednaka \(m\) dijagonala neke Hermitske idempotentne \(n \times n\) matrice. Upravo nas, prethodna varijanta dovodi i do posljednje trinaeste varijante Pitagorinog teorema, koja je teorem 3.2.4: Ako je \( \langle a_1, a_2, \dots a_n \rangle \) uređena n-torka realnih brojeva iz intervala [0, 1], čija je suma prirodan broj, onda postoji realna simetrična idempotentna \(n \times n\) matrica s dijagonalom \(a_1, a_2, \dots a_n \) |
| Sažetak (engleski) | In this work we have studied some of the variants of the Pythagorean theorem. After the history of Pythagoras, the Pythagorean theorem was proved in four different ways. The first evidence was through similarity of triangles. Followed by Euclid’s proof, Choupei’s and Garfield’s proof. Then some applications of the Pythagorean theorem in mathematics and physics were presented. Our first variant of Pythagorean theorem, shows that if we replace the two sides of the triangle by orthogonal axes and the hypotenuse by a vector \(x\) of the length \(c\) the Pythagorean theorem will be valid. Another variant, express \(x\) as the linear combination of certain unit vectors. The third variation is to, extend the previous statement on the finite dimensional unitary space of any dimension. Carpenter’s Theorem is our fourth variant and is inverse of the Pythagorean Theorem. The fifth variant is another formulation of the Pythagorean theorem in terms of the projections of vectors of equal length along the axes onto the line determined by a vector. The sixth and seventh variant (proposition 2.2.2 and proposition 2.2.3) gives us the answer to the question: Can something of this nature be said for orthonormal bases in higher-dimensional spaces? While the eighth variant (proposition 2.2.4) is a slightly reformulated seventh variant. The inverse of the eighth variant is our ninth variant. Tenth variant gives us the answer to the question whether there is a m-dimensional subspace of \(U\) on which projections of the basis elements have length of \(m\). In the last chapter, we are focused on variants of the Pythagorean theorem with operator-matrix methods. So, our eleventh variant indicates that the trace of a projection with m-dimensional range is equal to \(m\). The twelfth variant indicates that the ordered n-tuple \( \langle a_1, a_2, \dots a_n \rangle \) of numbers in [0, 1], with sum \(m\) is diagonal of some idempotent self-adjoint \(n \times n\) matrix. The previous variant leads to the last thirteenth variant of the Pythagorean theorem, which is a theorem 3.2.4: If \( \langle a_1, a_2, \dots a_n \rangle \) is an ordered n-tuple of numbers in [0, 1] with sum a positive integer, then there is an idempotent self-adjoint \(n \times n\) matrix with diagonal entries \(a_1, a_2, \dots a_n \)and all entries real. |
