Linear Algebra Recap Part 2 | Lecture Note

University:
University of California, Berkeley
Course:
MATH 56 Linear Algebra
Academic year:

2022
Views:

223

Pages:

11
Author:

Clint W.

Discussion 3 Vectors, Matrices and Inequalities Examples I Vectors: 1 x1 = , 2   1  x2 = 2 , 3 x> 1 = 1 2 , x> 2 = 1 2 3 . I Matrices: 1 2 3 A1 = , 0 0 1   1 0   A> 1 = 2 0 . 3 1 Discussion 3 Vectors, Matrices and Inequalities Examples: inner products I 2 × 3 matrix and 3-dimensional vector. 1×1+2×2+3×3 14 A1 x2 = = 0×1+0×2+1×3 3 I 3 × 2 matrix and 2-dimensional vector.     1×1+0×2 1    2 × 1 + 0 × 2 2 . A> x = = 1 1 3×1+1×2 5 I Some other products, e.g., x1> A1 , x2> A> 1. Discussion 3 Vectors, Matrices and Inequalities Examples: inner products Discussion 3 Vectors, Matrices and Inequalities Inequalities notation Consider x ∈ Rm , a1 , . . . , am ∈ Rn and [a1 . . . am ] = A ∈ Rn×m . The i-th component of aj is aij . Discussion 3 Vectors, Matrices and Inequalities Inequalities notation Consider x ∈ Rm , a1 , . . . , am ∈ Rn and [a1 . . . am ] = A ∈ Rn×m . The i-th component of aj is aij .    x1 = 1 .. x=1 means .   x = 1 m Discussion 3 Vectors, Matrices and Inequalities Inequalities notation Consider x ∈ Rm , a1 , . . . , am ∈ Rn and [a1 . . . am ] = A ∈ Rn×m . The i-th component of aj is aij .    x1 = 1 .. x=1 means .   x = 1 m    x1 < 0 .. x<0 means .   x < 0 m Discussion 3 Vectors, Matrices and Inequalities Inequalities notation Consider x ∈ Rm , a1 , . . . , am ∈ Rn and [a1 . . . am ] = A ∈ Rn×m . The i-th component of aj is aij .    x1 = 1 .. x=1 means .   x = 1 m    x1 < 0 .. x<0 means .   x < 0 m Ax ≥ 0 means P m   j=1 xj a1j ≥ 0  .. .   Pm x a ≥ 0 j=1 j nj Discussion 3 MaxMin Tricks Arbitrage idea The idea can be formulated as: max {min{Profit S1 , . . . , Profit S6 }} Can be re-written as: max z s.t. z ≤ Profit Si ∀i ∈ {1, . . . , 6} (1) Discussion 3 MaxMin Tricks General linearization of MaxMin (or MinMax) Given that g (y , x) is increasing with y for any x ∈ X , max g (min{f1 (x), . . . , fm (x)}, x) x∈X is equivalent (in some sense) to max g (z, x) x∈X s.t. z ≤ fi (x) ∀i ∈ {1, . . . , m} Discussion 3 MaxMin Tricks General linearization of MaxMin (or MinMax) Given that g (y , x) is increasing with y for any x ∈ X , max g (min{f1 (x), . . . , fm (x)}, x) x∈X is equivalent (in some sense) to max g (z, x) x∈X s.t. z ≤ fi (x) Proof by contradiction ∀i ∈ {1, . . . , m} Discussion 3 MaxMin Tricks General linearization of MaxMin (or MinMax) Similarly, for g (y , x) increasing with y , min g (max{f1 (x), . . . , fm (x)}, x) x∈X is equivalent (in some sense) to min g (z, x) x∈X s.t. z ≥ fi (x) ∀i ∈ {1, . . . , m}