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. Internal product
Multiply the elements at the same position and sum the multiplication results:
푣, 푤 ∈ ℝ푛 ∶
푤1 ⎡ 푤 ⎤ 푣푡 × 푤 = [푣 , 푣 , … , 푣 ] × ⎢ 2 ⎥ 1 2 푛 ⎢ … ⎥
⎣ 푤푛 ⎦ 푡 푛 푣 × 푤 = ∑푖=1 (푣푖 × 푤푖)
* the result of the internal product is a scalar. Vectors
Linear combination
Given 푚 vectors:
푣1, 푣2, … , 푣푚 ∈ ℝ푛. A linear combination of them is:
푚 1 2 푚 푖 휆1 × 푣 + 휆2 × 푣 + … + 휆푚 × 푣 = ∑ (휆푖 × 푣 ). 푖=1
Where, 휆1, 휆2, …, 휆푚 are scalars:
휆1, 휆2, … , 휆푚 ∈ ℝ. Vectors
Linear combination
1 2 푚 푚 푖 푣1 푣1 푣1 ∑푖=1 휆푖 × 푣1
1 2 푚 푚 푖 푣2 푣2 푣2 ∑푖=1 휆푖 × 푣2 휆1× + 휆2× + … + 휆푚× = … … … …
1 2 푚 푚 푖 푣푛 푣푛 푣푛 ∑푖=1 휆푖 × 푣푛 Vectors
Affine combination
If the sum of the scalars in a linear combination is equal to 1, then the combination is an AFFINE combination.
Affine combinations are linear combinations where:
푚
∑ 휆푖 = 1. 푖=1 Vectors
Conical combination
If the the scalars in a linear combination are non-negative, then the combination is a CONICAL combination.
Conical combinations are linear combinations where:
휆푖 ∈ ℝ+, ∀푖 ∈ {1, 2, … , 푚}. Vectors
Convex combination
If a linear combination is affine and conical, then the combination is a CONVEX combination.
Convex combinations are linear combinations where:
푚
∑ 휆푖 = 1 푖=1 휆푖 ∈ ℝ+, ∀푖 ∈ {1, 2, … , 푚}. Vectors
Linear subspace
푆 ⊆ ℝ푛 is a LINEAR SUBSPACE if for any two vectors 푣, 푤 ∈ 푆 and any two scalars 훼, 훽 ∈ ℝ, the vector resulting from the linear combination 훼 × 푣 + 훽 × 푤 is in 푆. Vectors
Affine subspace
푆 ⊆ ℝ푛 is an AFFINE SUBSPACE if for any two vectors 푣, 푤 ∈ 푆 and any two scalars 훼, 훽 ∈ ℝ, where 훼 + 훽 = 1, the vector resulting from the affine combination 훼 × 푣 + 훽 × 푤 is in 푆.
* Every linear subspace is an affine subspace. However, the opposite does not hold true. Vectors
Conical set
푆 ⊆ ℝ푛 is a CONICAL SET if for any two vectors 푣, 푤 ∈ 푆 and any two non-negative scalars 훼, 훽 ∈ ℝ+, the vector resulting from the conical combination 훼 × 푣 + 훽 × 푤 is in 푆.
* Every linear subspace is a conical set. However, the opposite does not hold true. Vectors
Convex set
푆 ⊆ ℝ푛 is a CONVEX SET if for any two vectors 푣, 푤 ∈ 푆 and any two non-negative scalars 훼, 훽 ∈ ℝ+, where 훼 + 훽 = 1, the vector resulting from the convex combination 훼 × 푣 + 훽 × 푤 is in 푆.
* Every conical, affine or linear subspace is a convex set. However, the opposite does not holdtrue. Vectors
Linear independence
The vectors 푣1, 푣2, … , 푣푚 ∈ ℝ푛 are LINEARLY INDEPENDENT if there exists only one linear combination resulting in the null vector, and is when all the scalars of the combination are equal to zero, i.e.:
푚 푖 푡 ∑ (휆푖 × 푣 ) = [0, 0, … , 0] ⇒ 휆푖 = 0, ∀푖 ∈ {1, 2, … , 푚} 푖=1 Vectors
Affine independence
The vectors 푣1, 푣2, … , 푣푚 ∈ ℝ푛 are AFFINELY INDEPENDENT if:
푚 푖 푡 ∑푖=1 (휆푖 × 푣 ) = [0, 0, … , 0] ⇒ 휆푖 = 0, ∀푖 ∈ {1, 2, … , 푚} 푚 ∑푖=1 휆푖 = 0
* Any set of linearly independent vectors is affine independent. However, the opposite does not hold true. Vectors
Linear span (hull)
The LINEAR SPAN (also called LINEAR HULL) of a set 푆 ⊆ ℝ푛 is the subspace containing all the linear combinations of the elements of 푆:
푚 푖 푖 푠푝푎푛(푆) = {∑ (휆푖 × 푣 ) | 푚 ∈ ℕ, 푣 ∈ 푆, 휆푖 ∈ ℝ} 푖=1 Vectors
Affine span (hull)
The AFFINE SPAN (also called AFFINE HULL) of a set 푆 ⊆ ℝ푛 is the subspace containing all the affine combinations of the elements of 푆:
푚 푚 푖 푖 푎푓푓(푆) = {∑ (휆푖 × 푣 ) | 푚 ∈ ℕ, 푣 ∈ 푆, 휆푖 ∈ ℝ, ∑ 휆푖 = 1} 푖=1 푖=0 Vectors
Conical hull
The CONICAL HULL of a set 푆 ⊆ ℝ푛 is the set containing all the conical combinations of the elements of 푆:
푚 푖 푖 푐표푛푒(푆) = {∑ (휆푖 × 푣 ) | 푚 ∈ ℕ, 푣 ∈ 푆, 휆푖 ∈ ℝ+} 푖=1 Vectors
Convex hull
The CONVEX HULL of a set 푆 ⊆ ℝ푛 is the set containing all the convex combinations of the elements of 푆:
푚 푚 푖 푖 푐표푛푣푒푥(푆) = {∑ (휆푖 × 푣 ) | 푚 ∈ ℕ, 푣 ∈ 푆, 휆푖 ∈ ℝ+, ∑ 휆푖 = 1} 푖=1 푖=0 Vectors
Basis
A set 푆 ⊂ ℝ푛 is a BASIS of ℝ푛, if the elements of 푆 are linearly independent and the linear span of 푆 is ℝ푛.
Remark. If 푆 is a basis of ℝ푛, then for any vector 푣 ∈ ℝ푛, there exists exactly one linear combination of the vectors of 푆 equals to 푣. Vectors
Change of basis
Consider a basis 푆 of ℝ푛 and a vector 푣 ∈ ℝ푛.
If the vector 푤 ∈ 푆 is multiplied by a scalar 휆 ≠ 0 in the linear combination of the elements of 푆 that results in 푣, then (푆 ⧵ {푤}) ∪ {푣} is a basis of ℝ푛. Matrices Matrices
Definition
Rectangular array of real values:
⎫ value 1, 1 value 1, 2 … value 1, 푛 } } } value 2, 1 value 2, 2 … value 2, 푛 } 푚×푛 + ⎬ 퐴푚×푛 ∈ ℝ ,(푚, 푛 ∈ ℕ ) … … … … } } value 푚, 1 value 푚, 2 … value 푚, 푛 } } ⎭ Matrices
Rows
푛 푎푖. ∈ ℝ is the 푖-th row of of the matrix 퐴푚×푛 (1 ≤ 푖 ≤ 푚):
value 1, 1 value 1, 2 … value 1, 푛
… … … …
value 푖, 1 value 푖, 2 … value 푖, 푛 푎푖. →
… … … …
value 푚, 1 value 푚, 2 … value 푚, 푛 Matrices
Columns
푚 푎.푗 ∈ ℝ is the 푗-th column of the matrix 퐴푚×푛 (1 ≤ 푗 ≤ 푛):
푎.푗 ↓
value 1, 1 … value 1, 푗 … value 1, 푛
value 2, 1 … value 2, 푗 … value 2, 푛
… … … … …
value 푚, 1 … value 푚, 푗 … value 푚, 푛 Matrices
Special matrices. Null
Each column of the matrix is a null vector (i.e. all elements are equal to zero):
0 … 0 … 0
… … … … …
0 … 0 … 0
… … … … …
0 … 0 … 0 Matrices
Special matrices. Identity
Square matrix (푚 = 푛), where 푎.푗 is the unit vector where the value 1 occurs at position 푗 (1 ≤ 푗 ≤ 푛) (i.e. all elements are equal to zero except for the main diagonal, whose elements are equal to one):
⎫ 1 0 0 … 0 } } 0 1 0 … 0 } } } 0 0 ⋱ ⋱ … 퐼 ∈ ℝ푛×푛 ⎬ 푛×푛 ⋱ ⋱ } … … 0 } } } 0 0 … 0 1 } ⎭ Matrices
Special matrices. Lower triangular
A square matrix is LOWER TRIANGULAR if all the values above the main diagonal are equal to zero (i.e. if 푖 < 푗, then 푎푖,푗 = 0):
푎1,1 0 0 … 0
푎2,1 푎2,2 0 … 0
⋱ 푎3,1 푎3,2 푎3,3 …
… … ⋱ ⋱ 0
푎푛,1 푎푛,2 … 푎푛,푛−1 푎푛,푛 Matrices
Special matrices. Upper triangular
A square matrix is UPPER TRIANGULAR if all the values below the main diagonal are equal to zero (i.e. if 푖 > 푗, then 푎푖,푗 = 0):
푎1,1 푎1,2 푎1,3 … 푎1,푛
0 푎2,2 푎2,3 … 푎2,푛
⋱ 0 0 푎3,3 …
⋱ ⋱ … … 푎푛−1,푛
0 0 … 0 푎푛,푛 Matrices
Operations. Scalar multiplication
Multiply each element of the matrix by the scalar:
⎫ } 휆 × 푎1,1 휆 × 푎1,2 … 휆 × 푎1,푛 } 푚×푛 } 휆 ∈ ℝ, 퐴푚×푛 ∈ ℝ ∶ } 휆 × 푎2,1 휆 × 푎2,2 … 휆 × 푎2,푛 } 휆 × 퐴 ∈ ℝ푚×푛 … … … … ⎬ } } 휆 × 푎푚,1휆 × 푎푚,2 … 휆×푎푚,푛 } } } ⎭ Matrices
Operations. Scalar sum
Sum each element of the matrix with the scalar:
⎫ } 휆 + 푎1,1 휆 + 푎1,2 … 휆 + 푎1,푛 } 푚×푛 } 휆 ∈ ℝ, 퐴푚×푛 ∈ ℝ ∶ } 휆 + 푎2,1 휆 + 푎2,2 … 휆 + 푎2,푛 } 휆 + 퐴 ∈ ℝ푚×푛 … … … … ⎬ } } 휆 + 푎푚,1휆 + 푎푚,2 … 휆+푎푚,푛 } } } ⎭ Matrices
Operations. Sum
Sum the elements at the same position (row and column):
⎫ } 푎1,1 + 푏1,1 푎1,2 + 푏1,2 … 푎1,푛 + 푏1,푛 } 푚×푛 }퐴푚×푛, 퐵푚×푛 ∈ ℝ ∶ } 푎2,1 + 푏2,1 푎2,2 + 푏2,2 … 푎2,푛 + 푏2,푛 } 퐴 + 퐵 ∈ ℝ푚×푛 … … … … ⎬ } } 푎푚,1 + 푏푚,1 푎푚,2 + 푏푚,2 … 푎푚,푛 + 푏푚,푛 } } } ⎭ Matrices
Operations. Multiplication
푚×푛 The matrix multiplication 퐴 × 퐵, where 퐴푚×푛 ∈ ℝ and 푝×푞 퐵푝×푞 ∈ ℝ , can be calculated iff 푛 = 푝 (i.e. the number of columns of the first matrix must be equal to the number of linesof the second one).
The resulting matrix 퐶 = 퐴 × 퐵 satisfies: 퐶 ∈ ℝ푚×푞 and the element 푐푖푗 is equal to the internal product of row 푎푖. of 퐴 by column 푏.푗 of 퐵 (1 ≤ 푖 ≤ 푚 and 1 ≤ 푗 ≤ 푞).
* Matrix multiplication is not a symmetric operation. Matrices
Operations. Multiplication
푚×푛 푛×푞 퐴푚×푛 ∈ ℝ , 퐵푛×푞 ∈ ℝ ∶
⎫ ∑푛 푎 × 푏 ∑푛 푎 × 푏 ∑푛 푎 × 푏 } 푖=1 1,푖 푖,1 푖=1 1,푖 푖,2 … 푖=1 1,푖 푖,푞 } } ∑푛 푎 × 푏 ∑푛 푎 × 푏 ∑푛 푎 × 푏 } 푖=1 2,푖 푖,1 푖=1 2,푖 푖,2 … 푖=1 2,푖 푖,푞 } 퐴 × 퐵 ∈ ℝ푚×푞 … … … … ⎬ } } ∑푛 푎 × 푏 ∑푛 푎 × 푏 ∑푛 푎 × 푏 푖=1 푚,푖 푖,1 푖=1 푚,푖 푖,2 … 푖=1 푚,푖 푖,푞 } } } ⎭ Matrices
Operations. Transpose
Given the matrix 퐴 ∈ ℝ푚×푛, the transpose is 퐴푡 ∈ ℝ푛×푚, where 푡 푡 the column 푎.푗 of 퐴 is equal to the row 푎푗. of 퐴 (1 ≤ 푗 ≤ 푛).
If 퐴 = 퐴푡 then 퐴 is SYMMETRIC and if 퐴 = −퐴푡 then 퐴 is ANTISYMMETRIC. Matrices
Inverse
A square matrix 퐴푛×푛 is INVERTIBLE (also known as non-singular or non-degenerate) if there exists a matrix 퐵푛×푛 (INVERSE), such that the product of 퐴 and 퐵 is equal to the identity:
퐴 × 퐵 = 퐼푛×푛
The INVERSE of 퐴 is denoted by 퐴−1.
* 퐴푛×푛 is invertible iff the lines of 퐴 are linearly independent. Matrices
Determinant
푛×푛 Each square matrix 퐴푛×푛 ∈ ℝ has associated a scalar number named DETERMINANT (푑푒푡(퐴) or |퐴|).
푛 푑푒푡(퐴) = |퐴| = ∑ 푠푖푔푛(휋) ∏ 푎 . 푖휋푖 휋∈풫푛 푖=1
Where, 풫푛 is the set of all permutations of the elements in {1, … , 푛} and 푠푖푔푛(휋) is equal to (−1)푘, being 푘 the number of two elements swap required to reorder 휋.
* The determinant of a 푛 × 푛 matrix can be computed within 풪 (푛2.373) complexity time. Matrices
Rank
푚×푛 The RANK of a matrix 퐴푚×푛 ∈ ℝ is an integer value equals to the maximum number of matrix rows that are linearly independent.
푟푎푛푘(퐴) = max |퐿|. 퐿 ⊆ {푎푖.|1 ≤ 푖 ≤ 푛} and 퐿 is linearly independent
* if 퐴 is a 푛 × 푛 matrix and 푑푒푡(퐴) ≠ 0, then 푟푎푛푘(퐴) = 푛. Linear systems Linear systems
Definition
푚×푛 A LINEAR SYSTEM receives a matrix 퐴푚×푛 ∈ ℝ and a vector 푏 ∈ ℝ푚, and seeks for a vector 푥 ∈ ℝ푛, such that 퐴 × 푥 = 푏.
퐴 푥 푏
푎1,1 푎1,2 … 푎1,푛 푥1 푏1
푎2,1 푎2,2 … 푎2,푛 푥2 푏2 × = … … … … … …
푎푚,1 푎푚,2 … 푎푚,푛 푥푛 푏푚 Linear systems
Elementary operations. Swap lines
푛 A vector 푥 ∈ ℝ is a solution of a linear system 퐴푚×푛 × 푥 = 푏 iff 푥 is a solution of 퐴′ × 푥 = 푏′, where 퐴′, 푏′ are obtained by swapping, respectively, the lines 푎푖. and 푎푗. in 퐴 and the entries 푏푖 and 푏푗 in 푏 (1 ≤ 푖, 푗 ≤ 푚). Linear systems
Elementary operations. Swap lines
Equivalent system obtained by swapping 푎푖. with 푎푗. and 푏푖 with 푏푗.
푎1,1 푎1,2 … 푎1,푛 푥1 푏1
… … … … 푥2 …
푎 푎 푎 푎푖. → 푖,1 푖,2 … 푖,푛 … 푏푖
… … … … × … = …
푎푗,1 푎푗,2 … 푎푗,푛 … 푏푗 푎푗. →
… … … … … …
푎푚,1 푎푚,2 … 푎푚,푛 푥푛 푏푚 Linear systems
Elementary operations. Scalar multiplication
푛 A vector 푥 ∈ ℝ is a solution of a linear system 퐴푚×푛 × 푥 = 푏 iff 푥 is a solution of 퐴′ × 푥 = 푏′, where 퐴′, 푏′ are obtained by multiplying the line 푎푖. in 퐴 and the entry 푏푖 in 푏 by a scalar 휆 ≠ 0 (1 ≤ 푖 ≤ 푛).
푎1,1 푎1,2 … 푎1,푛 푥1 푏1
… … … … 푥2 …
휆 × 푎 휆 × 푎 휆 × 푎 휆 × 푏 휆 × 푎푖. → 푖,1 푖,2 … 푖,푛 × … = 푖
… … … … … …
푎푚,1 푎푚,2 … 푎푚,푛 푥푛 푏푚 Linear systems
Elementary operations. Line substitution
푛 A vector 푥 ∈ ℝ is a solution of a linear system 퐴푚×푛 × 푥 = 푏 iff 푥 is a solution of 퐴′ × 푥 = 푏′, where 퐴′, 푏′ are obtained by replacing the line 푎푖. in 퐴 and the entry 푏푖 in 푏 respectively by 푎푖. + 휆푎푗. and 푏푖 + 휆푏푗 (1 ≤ 푖, 푗 ≤ 푛).
푎1,1 푎1,2 … 푎1,푛 푥1 푏1
… … … … 푥2 …
푎푖,1 + 푎푖,2 + 푎푖,푛 + … … 푏푖+휆×푏푗 푎푖. + 휆 × 푎푗. → 휆 × 푎푗,1 휆 × 푎푗,2 휆 × 푎푗,푛 × =
… … … … … …
푎푚,1 푎푚,2 … 푎푚,푛 푥푛 푏푚 Linear systems
Elementary operations. Permutations
푚×푚 A square matrix 푃푚×푚 ∈ ℝ is a PERMUTATION MATRIX if is obtained by the elementary operations over the identity matrix 퐼푚×푚.
푛 Given a permutation matrix 푃푚×푚, a vector 푥 ∈ ℝ is a solution ′ ′ of a linear system 퐴푚×푛 × 푥 = 푏 iff 푥 is a solution of 퐴 × 푥 = 푏 , where 퐴′ = 푃 × 퐴 and 푏′ = 푃 × 푏. Linear systems
Solving linear systems. Gauss method
Given a linear system 퐴 × 푥 = 푏, apply elementary operations to obtain an equivalent system 퐴′ × 푥 = 푏′, where 퐴′ is upper triangular and the elements of the diagonal are equal to 1. Then, use back substitution to find the entries of 푥. Linear systems
Solving linear systems. Inverse
Given a linear system 퐴 × 푥 = 푏 if 퐴 is invertible, then 푥 = 퐴−1 × 푏
−1 푡ℎ 퐴 × 푎 .푖 = 퐼.푖, where 퐼.푖 is the 푖 column of the identity. Thus, 푡ℎ −1 the 푖 of 퐴 can be computed by solving 퐴 × 푥 = 퐼.푖. Linear systems
Solving linear systems. Cramer’s rule
Given a linear system 퐴 × 푥 = 푏 if 퐴 is invertible, then 푥 = 푑푒푡(퐵푖) 퐵 퐴 푖 푑푒푡(퐴) , where 푖 is obtained by replacing in the column 푎.푖 with b. Linear systems
Solving linear systems
Given a linear system 퐴푚×푛 × 푥 = 푏, consider the matrix (퐴, 푏)푚×(푛+1), whose first 푚 rows are the rows from 퐴 and the last row is the vector 푏. Then:
▶ If 푟푎푛푘(퐴, 푏) > 푟푎푛푘(퐴), then the linear system has no solution. ▶ If 푟푎푛푘(퐴, 푏) = 푟푎푛푘(퐴) = 푛, then the linear system has exactly one solution. ▶ If 푟푎푛푘(퐴, 푏) = 푟푎푛푘(퐴) < 푛, then the linear system has an infinite number of solutions Exercises Exercises
Exercise 1.
Given 푆 ⊆ ℝ푛 demonstrate that:
푆 is a linear subspace ⇒ 푆 is an affine subspace. Exercises
Exercise 2.
Demonstrate that:
If 푆 ⊆ ℝ푛 is a convex set, not necessarily 푆 is an affine subspace. Exercises
Exercise 3.
Given the set of vector 푆 ⊆ ℝ푛 Demonstrate that:
▶ If the vectors of 푆 are linearly independent, then they are also affine independent.
▶ If the vector of 푆 are affine independent, not necessarily they are linear independent. Exercises
Exercise 4.
Demonstrate that:
If 푆 ⊆ ℝ푛 is a basis of ℝ푛, then for any vector 푣 ∈ ℝ푛 there exists exactly one linear combination of the vectors of 푆 equals to 푣. Exercises
Exercise 5.
Demonstrate the change of basis property. Exercises
Exercise 6.
Suppose are given two matrices 퐴 and 퐵, such that it is possibly to multiply 퐴 × 퐵 and also 퐵 × 퐴. Demonstrate that not necessarily 퐴 × 퐵 = 퐵 × 퐴. Exercises
Exercise 7.
Demonstrate that:
▶ (퐴푡)푡 = 퐴.
▶ (퐴 + 퐵)푡 = 퐴푡 + 퐵푡.
▶ (퐴 × 퐵)푡 = 퐵푡 × 퐴푡. Exercises
Exercise 8.
Given an invertible matrix 퐴 ∈ ℝ푛×푛, demonstrate that:
−1 ▶ 퐴−1 is unique, invertible and (퐴−1) = 퐴.
푡 ▶ 퐴푡 is invertible and (퐴푡)−1 = (퐴−1) .
▶ If 퐵 ∈ ℝ푛×푛 is invertible, then 퐴 × 퐵 is invertible and (퐴 × 퐵)−1 = 퐵−1 × 퐴−1. Exercises
Exercise 9.
Demonstrate that:
A square matrix 퐴 is invertible iff 푑푒푡(퐴) ≠ 0. Linear algebra
Integer linear programming: formulations, techniques and applications.
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Santiago Valdés Ravelo [email protected] August, 2020