| SM ISO690:2012|
POPA, Alexandru. Orthonormalization of limit lineal basis. In: Mathematics and IT: Research and Education, 1-3 iulie 2021, Chişinău. Chișinău, Republica Moldova: 2021, pp. 71-72.
|Mathematics and IT: Research and Education 2021|
Conferința "Mathematics and IT: Research and Education " |
Chişinău, Moldova, 1-3 iulie 2021
Let Bn be some homogeneous space  with specification fk1; :::; kng, kp 2 f¡1; 0; 1g; p = 1; n. Define:formulaFor not null vector x define index as one of (smallest) index i so that x¯i x > 0. The notation x ¯ y = x ¯min(i;j) y, where i is index of x and j is index of y. If there is no such index, x ¯i x = 0; 8i = 0; n, then vector x is named limit. As stated in , limit vector always can be presented as sum of indexed vectors x = a + b, so that a ? b; a ¯ a = b ¯ b. Vectors a; b are named decomposition vectors of vector x. Lemmas 1 and 2 from  state for orthogonal vectors, from which at least one is limit, the possibility to orthogonalize also their decomposition vectors. Lemma 1. Indices of decomposition vectors do not depend on space basis choice. Define lineal as linear span of vectors (it may be congruent or not with some subspace of Bn). Define limit lineal as lineal whose orthonormal basis contains limit vectors. Lemma 2. In limit lineal basis, the indices of decomposition vectors are not among indices of indexed vectors. Lemmas 1 and 2 permit to consider decomposition vectors indices as double index of a limit vector. Remark 3. If not collinear limit vectors x; y are not orthogonal, then vectors x + y; x ¡ y are not collinear and indexed. Proposition 4. If limit vector x is not orthogonal with indexed vector y, it is possible to find decomposition vectors x = a + b such that either a or b is collinear with y. It is possible to develop the Algorithm of orthonormalization of vector family containing limit vectors, where decomposition vectors are normalized and orthogonal to all other vectors. Lemma 5. Orthogonal complement do not exist for limit lineals.Lemma 6. Generally speaking, it is not possible to project a vector onto limit lineal.