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Two.I Vector Space Definition Linear Algebra Jim Hefferon
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Definition and examples
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Vector space 1.1 Definition A vector space (over R ) consists of a set V along with two operations ‘+’ and ‘ · ’ subject to the conditions that for all vectors ~ v, ~ w, ~ u V , and all scalars r, s R : 1) the set V is closed under vector addition, that is, ~ v + ~ w V 2) vector addition is commutative ~ v + ~ w = ~ w + ~ v 3) vector addition is associative ( ~ v + ~ w ) + ~ u = ~ v + ( ~ w + ~ u ) 4) there is a zero vector ~ 0 V such that ~ v + ~ 0 = ~ v for all ~ v V 5) each ~ v V has an additive inverse ~ w V such that ~ w + ~ v = ~ 0
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Vector space 1.1 Definition A vector space (over R ) consists of a set V along with two operations ‘+’ and ‘ · ’ subject to the conditions that for all vectors ~ v, ~ w, ~ u V , and all scalars r, s R : 1) the set V is closed under vector addition, that is, ~ v + ~ w V 2) vector addition is commutative ~ v + ~ w = ~ w + ~ v 3) vector addition is associative ( ~ v + ~ w ) + ~ u = ~ v + ( ~ w + ~ u ) 4) there is a zero vector ~ 0 V such that ~ v + ~ 0 = ~ v for all ~ v V 5) each ~ v V has an additive inverse ~ w V such that ~ w + ~ v = ~ 0 6) the set V is closed under scalar multiplication, that is, r · ~ v V 7) addition of scalars distributes over scalar multiplication ( r + s ) · ~ v = r · ~ v + s · ~ v 8) scalar multiplication distributes over vector addition r · ( ~ v + ~ w ) = r · ~ v + r · ~ w 9) ordinary multipication of scalars associates with scalar multiplication ( rs ) · ~ v = r · ( s · ~ v ) 10) multiplication by the scalar 1 is the identity operation 1 · ~ v = ~ v .
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Example Consider the set of row vectors consisting of all multiples of ( 1 2 ) . V = { ( a 2a ) | a R } Some members of V are ( 4 8 ) , ( 1/2 1 ) , (- 100 - 200 ) , and ( 0 0 ) .
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Example Consider the set of row vectors consisting of all multiples of ( 1 2 ) . V = { ( a 2a ) | a R } Some members of V are ( 4 8 ) , ( 1/2 1 ) , (- 100 - 200 ) , and ( 0 0 ) . This V is a vector space under the natural addition ( a 1 2a 1 ) + ( a 2 2a 2 ) = ( a 1 + a 2 2a 1 + 2a 2 ) and scalar multiplication operations. r ( a 1 2a 1 ) = ( ra 1 2ra 1 ) To verify that, we will check each of the ten conditions. Because this is the first time through the definition, we will verify these at length.
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We first check closure under addition (1), that the sum of two members of V is also a member of V . Take ~ v and ~ w to be members of V . ~ v = ( v 1 2v 1 ) ~ w = ( w 1 2w 1 ) Then their sum ~ v + ~ w = ( v 1 + w 1 2v 1 + 2w 1 ) is also a member of V because its second entry is twice its first.
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We first check closure under addition (1), that the sum of two members of V is also a member of V . Take ~ v and ~ w to be members of V . ~ v = ( v 1 2v 1 ) ~ w = ( w 1 2w 1 ) Then their sum ~ v + ~ w = ( v 1 + w 1 2v 1 + 2w 1 ) is also a member of V because its second entry is twice its first. Condition (2), commutativity of addition, is straightforward. The sums in the two orders are ~ v + ~ w = ( v 1 + w 1 2 ( v 1 + w 1 )) and ~ w + ~ v = ( w 1 + v 1 2 ( w 1 + v 1 )) and the two are equal because v 1 + w 1 equals w 1 + v 1 , as both are sums of real numbers and real number addition is commutative.
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Condition (3), associativity of addition, is like the prior one. The left side is ( ~ v + ~ w ) + ~ u = (( v 1 + w 1 ) + u 1 ( 2v 1 + 2w 1 ) + 2u 1 ) while the right side is this.
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