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Affine Spaces Def: Suppose

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1 Affine Spaces Def: Suppose 𝑎 1 ,…, 𝑎 𝑘 ∈ 𝑅 𝑛 . Then 𝑖=1 𝑘 𝜆 𝑖 𝑎 𝑖 , where 𝜆 𝑖 ∈𝑅, is an affine combination of the 𝑎 𝑖 ′ 𝑠 when 𝑖=1 𝑘 𝜆 𝑖 =1. Def: Affine space in 𝑅 𝑛 : subset 𝐿⊆ 𝑅 𝑛 closed under affine combinations, i.e. 𝑎 1 ,…, 𝑎 𝑘 ∈𝐿⊆ 𝑅 𝑛 , 𝜆 1 ,…, 𝜆 𝑘 ∈𝑅, 𝑖=1 𝑘 𝜆 𝑖 =1  𝑖=1 𝑘 𝜆 𝑖 𝑎 𝑖 ∈𝐿 Ex) 𝐿={𝑥∈ 𝑅 𝑛 :𝐴𝑥=𝑏}, where 𝐴:𝑚×𝑛, 𝑏∈ 𝑅 𝑚 . Then L is an affine space. (Let 𝑥 1 ,…, 𝑥 𝑘 ∈𝐿, i.e. 𝐴 𝑥 𝑖 =𝑏 and suppose 𝜆 𝑖 , 1≤𝑖≤𝑘 satisfy 𝑖=1 𝑘 𝜆 𝑖 =1. Consider 𝑥≡ 𝑖=1 𝑘 𝜆 𝑖 𝑥 𝑖 . Then 𝐴𝑥=𝐴( 𝑖=1 𝑘 𝜆 𝑖 𝑥 𝑖 ) = 𝑖=1 𝑘 𝜆 𝑖 𝐴 𝑥 𝑖 = 𝑖=1 𝑘 𝜆 𝑖 𝑏 =𝑏, i.e. 𝐿 is closed under affine combinations. ) Linear Programming 2016

2 Affine Spaces Affine combination of 𝑥,𝑦∈ 𝑅 2 𝑥 𝑦
Linear Programming 2016

3 Let 𝑆⊆ 𝑅 𝑛 be a subspace and 𝑎∈ 𝑅 𝑛 .
𝐿=𝑆+ 𝑎 ≡{𝑥+𝑎: 𝑥∈𝑆} is a translation of subspace 𝑆. 𝐿 is then an affine space. ( 𝑖=1 𝑘 𝜆 𝑖 𝑥 𝑖 +𝑎 = 𝑖=1 𝑘 𝜆 𝑖 𝑥 𝑖 + 𝑖=1 𝑘 𝜆 𝑖 𝑎 = 𝑖=1 𝑘 𝜆 𝑖 𝑥 𝑖 +𝑎 ∈𝐿) Thm: Let 𝐿⊆ 𝑅 𝑛 . Then 𝐿 is an affine space  𝐿 is a translation of a subspace 𝑆 in 𝑅 𝑛 . (𝐿=𝑆+ 𝑎 , 𝑎∈𝐿) Pf) ( ) above (  ) HW later.  Analogue to generation in case of subspaces, i.e. 𝐿 can be affinely generated (finitely) by vectors { 𝑎 1 +𝑎,…, 𝑎 𝑘 +𝑎,𝑎}, where { 𝑎 1 ,…, 𝑎 𝑘 } is a basis for 𝑆 with 𝐿=𝑆+{𝑎}. (𝐿= 𝑖=1 𝑘 𝜆 𝑖 𝑎 𝑖 +𝑎, 𝑎∈𝐿 = 𝑖=1 𝑘 𝜆 𝑖 𝑎 𝑖 + 𝑖=1 𝑘 𝜆 𝑖 𝑎 − 𝑖=1 𝑘 𝜆 𝑖 𝑎 +𝑎, 𝑎∈𝐿 𝑖=1 𝑘 𝜆 𝑖 𝑎 𝑖 + 𝑖=1 𝑘 𝜆 𝑖 𝑎 − 𝑖=1 𝑘 𝜆 𝑖 𝑎 +𝑎, 𝑎∈𝐿 ={ 𝑖=1 𝑘 𝜆 𝑖 𝑎 𝑖 +𝑎 + 1− 𝑖=1 𝑘 𝜆 𝑖 𝑎, 𝑎∈𝐿} , sum of weights is 1.) Linear Programming 2016

4 Generation of Affine Space
𝐿=𝑆+{𝑎} 𝐿=𝑆+{𝑎} 𝑎+ 𝑎 2 𝑎+ 𝑎 1 𝑎 𝑎 2 𝑎 1 𝑆 Linear Programming 2016

5 𝐿=𝑆+{𝑎} ={𝑥:𝑥=𝑧+𝑎, and 𝑧∈𝑆} ={𝑥:𝑥=𝑧+𝑎, and 𝐴𝑧=0}
Thm: Let 𝐿⊆ 𝑅 𝑛 . Then 𝐿 is an affine space  𝐿={𝑥:𝐴𝑥=𝑏} for some 𝐴:𝑚×𝑛, 𝑏∈ 𝑅 𝑚 . Pf) (  ) already given. (  ) By previous theorem, we have 𝐿=𝑆+{𝑎}, where 𝑆 is a subspace of 𝑅 𝑛 . (Note that we may assume 𝑆≠∅, because argument true when 𝐿=∅ by taking 𝐴= 0,…,0 , 𝑏=[1].) Thus 𝑆 is a constrained subspace (from previous result), i.e. 𝑆={𝑧:𝐴𝑧=0} for some matrix 𝐴. 𝐿=𝑆+{𝑎} ={𝑥:𝑥=𝑧+𝑎, and 𝑧∈𝑆} ={𝑥:𝑥=𝑧+𝑎, and 𝐴𝑧=0} = 𝑥:𝐴 𝑥−𝑎 =0 ={𝑥:𝐴𝑥= 𝐴𝑎 } Hence 𝐿 is a solution set for finite system of linear equations.  Linear Programming 2016

6 These are the same for any 𝐴⊆ 𝑅 𝑛 .
Def: Affine span of 𝐴⊆ 𝑅 𝑛 is the set of all affine combinations of points of 𝐴 (denoted 𝐿(𝐴)) Affine hull of 𝐴⊆ 𝑅 𝑛 is 𝑖∈𝐼 𝐿 𝑖 , 𝐿 𝑖 ⊇𝐴, 𝐿 𝑖 ⊆ 𝑅 𝑛 , 𝑖∈𝐼, and affine space ( 𝐼 is index set of affine spaces containing 𝐴). ( Recall linear hull of 𝐴⊆ 𝑅 𝑛 is “the smallest subspace containing 𝐴, i.e., linear hull of 𝐴= 𝑖 𝑆 𝑖 , 𝑆 𝑖 ⊇𝐴, 𝑆 𝑖 subspace in 𝑅 𝑛 ). These are the same for any 𝐴⊆ 𝑅 𝑛 . Def: Set of points 𝑎 1 ,…, 𝑎 𝑚 ∈ 𝑅 𝑛 are affinely dependent when some 𝑎 𝑖 is an affine combination of remaining 𝑎 𝑗 , 𝑗≠𝑖. Otherwise, affinely independent. Linear Programming 2016

7 Affine dependence Linear Programming 2016

8 Affine independence Linear Programming 2016

9 Thm: (i) 𝑎 1 ,…, 𝑎 𝑚 affinely independent
 𝑖=1 𝑚 𝜆 𝑖 𝑎 𝑖 =0, 𝑖=1 𝑚 𝜆 𝑖 =0 implies 𝜆 𝑖 =0 for all 𝑖. (ii) 𝑎 1 ,…, 𝑎 𝑚 affinely independent  for each 𝑘, the vectors 𝑎 𝑖 − 𝑎 𝑘 , ∀𝑖≠𝑘 are linearly independent. (iii) 𝑎 1 ,…, 𝑎 𝑚 affinely independent  𝑎 1 ,−1 ,…,( 𝑎 𝑚 ,−1)∈ 𝑅 𝑛+1 are linearly independent. Pf) HW. Note that linear independence of a set of vectors implies affine independence, but converse is not true. Linear Programming 2016

10 Equivalent conditions for affine independence (ii)
𝑎 1 −𝑎 𝑎 1 𝑎 𝑎 2 −𝑎 𝑎 2 Linear Programming 2016

11 Equivalent conditions for affine independence (iii)
𝑎 1 𝑎 2 𝑎 3 ( 𝑎 1 ,−1) ( 𝑎 2 −1) -1 ( 𝑎 3 ,−1) Linear Programming 2016

12 Def: For affine space 𝐿, define dimension of 𝐿 as dim(𝐿) = dim(𝑆), where 𝐿=𝑆+{𝑎}.
For arbitrary subset 𝐴⊆ 𝑅 𝑛 , define dimension of 𝐴 as dim(𝐴) = dim(𝐿(𝐴)). Finally, affine rank of a set 𝐴 is the size of a largest affinely independent subset of 𝐴. Linear Programming 2016

13 Thm: Suppose 𝐴⊆ 𝑅 𝑛 and dim(𝐴) = 𝑘. Then 𝐴 has affine rank (𝑘+1).
Pf) Suppose 𝐿 𝐴 =𝑆+{𝑎} with dim(𝑆) = 𝑘. 𝐴 contains at most 𝑘+1 affinely independent vectors. Else if 𝑎 0 , 𝑎 1 ,…, 𝑎 𝑘 , 𝑎 𝑘+1 affinely independent vectors in 𝐴, then 𝑎 1 − 𝑎 0 ,…, 𝑎 𝑘+1 − 𝑎 0 is a linearly independent set of vectors in 𝑆, contradiction to dim(𝑆) = 𝑘. Also 𝐴 contains at least 𝑘+1 affinely independent vectors. 𝑆 is generated by vectors of form 𝑦 𝑖 −𝑎, where 𝑦 𝑖 ,𝑎∈𝐴. ( 𝐿 𝐴 = 𝑖=1 𝑝−1 𝜆 𝑖 𝑎 𝑖 + 𝜆 𝑝 𝑎 𝑝 , 𝑖=1 𝑝 𝜆 𝑖 =1 , where 𝑎 𝑖 ∈𝐴, 𝑖=1,…,𝑝 = 𝑖=1 𝑝−1 𝜆 𝑖 𝑎 𝑖 +(1− 𝑖=1 𝑝−1 𝜆 𝑖 ) 𝑎 𝑝 = 𝑎 𝑝 + 𝑖=1 𝑝−1 𝜆 𝑖 𝑎 𝑖 − 𝑎 𝑝 = 𝑎 𝑝 +𝑆 ) But 𝑆 has dim(𝑆) = 𝑘, so  points 𝑦 1 −𝑎,…, 𝑦 𝑘 −𝑎, basis for 𝑆, hence 𝑎, 𝑦 1 ,…, 𝑦 𝑘 are affinely independent members of 𝐴.  Linear Programming 2016


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