Linear Combinations in Real Linear Space

Wojciech A. Trybulec

Warsaw University

Summary.

The article is continuation of [17]. At the beginning we prove some theorems concerning sums of finite sequence of vectors. We introduce the following notions: sum of finite subset of vectors, linear combination, carrier of linear combination, linear combination of elements of a given set of vectors, sum of linear combination. We also show that the set of linear combinations is a real linear space. At the end of article we prove some auxiliary theorems that should be proved in [8], [5], [9], [2] or [10].

MML Identifier: RLVECT_2

Contents (PDF format)

Bibliography

[1] Grzegorz Bancerek. Cardinal numbers. Journal of Formalized Mathematics, 1, 1989.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.

[4] Czeslaw Bylinski. Binary operations. Journal of Formalized Mathematics, 1, 1989.

[5] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[6] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.

[7] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.

[8] Library Committee. Boolean properties of sets --- requirements. Journal of Formalized Mathematics, EMM, 2002.

[9] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.

[10] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.

[11] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.

[12] Andrzej Trybulec. Enumerated sets. Journal of Formalized Mathematics, 1, 1989.

[13] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[14] Andrzej Trybulec. Function domains and Fr\aenkel operator. Journal of Formalized Mathematics, 2, 1990.

[15] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.

[16] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Journal of Formalized Mathematics, 1, 1989.

[17] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.

[18] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.

[19] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

[20] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.