2.2 Basis and dimension vector spaces - Basis and dimension...

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Basis and dimension vector spaces. Definition 1. If S ={ v 1 ,v 2 ,…, v k } is a set of vectors in a vector space V , then the set of all vectors in vector space V that are linear combination of the vectors in S is denoted by the span S or span { v 1 ,v 2 ,…, v k } . Note that the span of any nonempty system of vectors is a vector space. Example 0. 1. Span{(0,1)}={c(0,1)=(0,c): c is any number}. In plane it is y- axes. 2. Span{(1,0)}={c(1,0)=(c,0): c is any number}. In plane it is x- axes. 3. Span{(1,1)}={c(1,1)=(c,c): c is any number}. In plane it is the straight line y=x. Definition 2. The vectors v 1 ,v 2 ,…,v k in vector space V a said to form a basis for V if a). v 1 ,v 2 ,…,v k spanV and b). v 1 ,v 2 ,…,v k are linearly independent. Characterizations of a Basis Let S ={ v 1 ,v 2 ,…, v k } be a set of vectors in a vector space V. The following statements are equivalent. 1) S is a basis for V. 2 ¿ S is a minimal spanning set for V. 3) S is a maximal linearly independent subset of V.
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Example 1. The vectors e 1 =( 1,0 ) end e 2 =( 0,1 ) form a basis for R 2 , because a). if a =( a 1 ,a 2 ) is any vector from R 2 , then a = ( a 1 ,a 2 ) = a 1 ( 1,0 ) + a 2 ( 0,1 ) = a 1 e 1 + a 2 e 2 . That means that any vector from R 2 is linear combination of the vectors e 1 and e 2 .
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