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# lecture_25 - MA 265 LECTURE NOTES FRIDAY MARCH 7 Bases and...

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MA 265 LECTURE NOTES: FRIDAY, MARCH 7 Bases and Dimension Basic Results. Let ( V, + , · ) be a real vector space of dimension n . We recall a few statements we made without proof at the end of the previous lecture. (1) If T is a maximal independent set of vectors in V , then T contains exactly n vectors. (2) If T is a set of n linearly independent vectors, then it is a basis for V . (3) Any collection S of k > n vectors must be linearly dependent. (4) If T is a minimal spanning set for V , then T contains exactly n vectors. (5) It T is a set of n vectors which spans V , then it is a basis for V . (6) Any collection of S of k < n vectors cannot span V . We explain why these are true. For (1), let T = { v 1 , v 2 , . . . , v m } be an maximal independent set of vectors. We show that T is a basis for V . If this were not true, then T would not span V , so that we could find some vector v V not in the span of T . Consider S = { v 1 , v 2 , . . . , v m , v } . If S were linearly dependent, then we may write a 1 v 1 + a 2 v 2 + · · · + a m v m + a v = 0 = v = - a 1 a v 1 + - a 2 a v 2 + · · · + - a m a v m so that v is in the span of T – a contradiction. Hence S must be a linearly independent set which properly contains T . This contradicts the maximality of T – so that our original assumption on T must have been incorrect. Hence T must be a basis for

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lecture_25 - MA 265 LECTURE NOTES FRIDAY MARCH 7 Bases and...

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