two_i_allproofs.pdf

# two_i_allproofs.pdf - Two.I Vector Space Definition Linear...

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Two.I Vector Space Definition Linear Algebra Jim Hefferon

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Definition and examples
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 .
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.
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.
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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• Fall '16
• Darij Grinberg

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