# I-Soccer & Linear Algebra0417

Lapho umdlalo webhola lezinyawo uqala, ibhola lilele maphakathi nenkundla bese lihanjiswa lizungeze inkundla imizuzu engama-45 ngokushintsha nokujika. Ekuqaleni kwesiwombe sesibili ibhola liphinde libe maphakathi nenkundla. Sikhombisa ngezindlela ezilula ze-algebra eqondile ukuthi inani elingenamkhawulo lamaphuzu ebusweni lihlala lisesimweni esifanayo nasesimweni sokuqala noma ngqo 2.

Okokuqala, ukususwa kwebhola, okwenziwa phakathi nengxenye yokuqala, kuhlanganisa kancane kwi-vector zero. Ngakho-ke banganganakwa. Lokhu kushiya inani elilinganiselwe lokujikeleza $$A_1, ..., A_n \in \mathbb{R}^{3 \times 3}$$ nge $$A$$ orthogonal kanye $$\det(A_k) = 1 \,\, \forall \,\, k \in \{1,...,n\}$$ . $$A_i, A_j$$ kuyasebenza:

$$(A_i A_j)^T (A_i A_j) = A_j^T A_i^T A_i A_j = A_j^T (A_i^T A_i) A_j = A_j^T E_3 A_j = A_j^T A_j = E_3$$

njengoba

$$\det(A_i A_j) = \det(A_i) \cdot \det(A_j)=1 \cdot 1 = 1.$$

Lokhu kusho ukuthi $$A_i A_j$$ futhi ukujikeleza, yingakho $$A_1 ... A_n$$ nakho ukujikeleza okukodwa.

Uma manje $$A_1 ... A_n = E_n$$ , kusobala ukuthi wonke amaphuzu obuso bebhola asendaweni yokuqala - kwelinye (okungenzeka) icala i-eigenvector ye $$A_1 ... A_n$$ ilingana ne-axis yayo yokujikeleza inani le-eigenvalue $$1$$ . Lokhu kusho ukuthi impela lawa maphuzu amabili, asezingeni lokujikeleza, ahlelwe kuwo uqobo.

Emuva