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. )
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2. Affine Spaces Affine combination of 𝑥,𝑦∈ 𝑅 2 𝑥 𝑦
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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.)
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4. Generation of Affine Space
𝐿=𝑆+{𝑎}
𝐿=𝑆+{𝑎}
𝑎+ 𝑎 2
𝑎+ 𝑎 1 𝑎
𝑎 2
𝑎 1 𝑆
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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. 
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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.
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7. Affine dependence
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8. Affine independence
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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.
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10. Equivalent conditions for affine independence (ii)
𝑎 1 −𝑎
𝑎 1 𝑎
𝑎 2 −𝑎
𝑎 2
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11. Equivalent conditions for affine independence (iii)
𝑎 1
𝑎 2
𝑎 3
( 𝑎 1 ,−1)
( 𝑎 2 −1) -1
( 𝑎 3 ,−1)
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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.
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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