combinepdf-min-0348.pdf

# combinepdf-min-0348.pdf - Notes on linear algebra(Monday...

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Unformatted text preview: Notes on linear algebra (Monday 17th October. 2016, 23:10) page 195 4.3. <DRAFT> (*) The summation sign for vectors In Section 2.9,. we have introduced the summation sign 2 for sums of numbers; in Section 2.14, we have introduced the analogous summation sign 2 for sums of matrices. In the same way, we can define the summation sign 2 for sums of vectors in an arbitrary vector space: Deﬁnition 4.17. Let V be a vector space. Let p and a be two integers such that q p g a + 1. Let op, op“, . . .. vq be some vectors in V. Then. E ’01.: means the sum k=a "Up + vp+1 —|- - ' - —J— vq. (This sum is well-defined, since each vector space satisfies a ”general associativity law” similar to Proposition 2.23. Moreover, the order of its addends does not matter, as can be proven using property (a) in Definition 4.2.) q q The notation 2, ok is analogous to the notation Z ak introduced in Definition kzp kzp q 2.26 (and to the notation E Ak introduced in Definition 2.43). The same remarks k=p and clarifications done for the latter notation in Definition 2.27 apply to the q former notation. There is only one difference: When the sum 2 Uk has no k=P _; ...
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