Linear algebra Integer linear programming: formulations, techniques and applications. 03 Santiago Valdés Ravelo
[email protected] August, 2020 “Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.”1 Michael F. Atiyah. 1Michael F. Atiyah, “Special Article: Mathematics in the 20푡ℎ Century”, Bulletin of the London Mathematical Society, 2002, p. 7. 1 Vectors 2 Matrices 3 Linear systems 4 Exercises Vectors Vectors Definition Finite sequence of real values: ⎫ value 1 } } } value 2 } } } value 3 } 푛 + ⎬ 푣 ∈ ℝ , where 푛 ∈ ℕ … } } value 푛 − 1 } } } value 푛 } } ⎭ Vectors Special vectors. Null Vector with all the elements equal to zero (null): ⎫ 0 } } 0 } } 푣 = [0, 0, … , 0]푡 } … 푣 ⎬ } 0 } } } 0 } ⎭ Vectors Special vectors. Unit Vector with one element equals to 1 and the rest equal to 0: ⎫ 0 } } } … } } 0 } } 푣 = [0, … , 0, 1, 0, … , 0]푡 } 1 ⎬ 푣 } 0 } } } … } } 0 } } * unit vectors are those whose norm is equals to 1 ⎭ Vectors Operations. Sum Sum the elements at the same position: ⎫ } 푛 푣1 + 푤1 } 푣, 푤 ∈ ℝ ∶ } } 푣2 + 푤2 } 푣 + 푤 = [푣 , 푣 , … , 푣 ]푡 + [푤 , 푤 , … , 푤 ]푡 푣 + 푤 1 2 푛 1 2 푛 … ⎬ } } 푡 푣푛 + 푤푛 } 푣 + 푤 = [푣1 + 푤1, 푣2 + 푤2, … , 푣푛 + 푤푛] } } ⎭ Vectors Operations. Scalar multiplication Multiply each element of the vector by the scalar (real number): ⎫ } 푛 휆 × 푣1 } 휆 ∈ ℝ, 푣 ∈ ℝ ∶ } } 휆 × 푣2 } 휆 × 푣 = 휆 × [푣 , 푣 , … , 푣 ]푡 휆 × 푣 1 2 푛 … ⎬ } } 푡 휆 × 푣푛 } 휆 × 푣 = [휆 × 푣1, 휆 × 푣2, … , 휆 × 푣푛] } } ⎭ Vectors Operations.