Shawn Zhong
钟万祥
Toggle navigation
Tutorials
Mathematics
Math 240
Math 375
Math 431
Math 514
Math 521
Math 541
Math 632
Abstract Algebra
Linear Algebra
Category Theory
Computer Sciences
CS/ECE 252
CS/ECE 352
Learn Haskell
AP Notes
AP Microecon
AP Macroecon
AP Statistics
AP Chemistry
AP Physics E&M
AP Physics Mech
CLEP Psycho
Shawn Zhong
钟万祥
Toggle navigation
Tutorials
Mathematics
Math 240
Math 375
Math 431
Math 514
Math 521
Math 541
Math 632
Abstract Algebra
Linear Algebra
Category Theory
Computer Sciences
CS/ECE 252
CS/ECE 352
Learn Haskell
AP Notes
AP Microecon
AP Macroecon
AP Statistics
AP Chemistry
AP Physics E&M
AP Physics Mech
CLEP Psycho
Home
/
2017
/
November
/
3
Math 375 – 11/2
Nov 03, 2017
Shawn
Math 375
No comments yet
Search
Search for:
Home Page
Tutorials
Mathematics
Math 240 – Discrete Math
Math 375 – Linear Algebra
Math 431 – Intro to Probability
Math 514 – Numerical Analysis
Math 521 – Analysis I
Math 541 – Abstract Algebra
Math 632 – Stochastic Processes
Abstract Algebra @ 万门大学
Linear Algebra @ 万门大学
Category Theory
Computer Sciences
CS/ECE 252 – Intro to Computer Engr.
CS/ECE 352 – Digital System Fund.
Learn Haskell
Course Notes
AP Macroeconomics
AP Microeconomics
AP Chemistry
AP Statistics
AP Physics C: E＆M
AP Physics C: Mechanics
CLEP Psychology
2048 Game
HiMCM 2016
登峰杯 MCM
WeChat Account
TOP
![Uniqueness Theorem • Theorem ○ Suppose f(A_1,…,A_n ) is a function of A_1,…,A_n∈Rn ○ That satisfies Linearity and Alternating § f(B+C,A_2,…,A_n )=f(B,A_2,…,A_n )+f(C,A_2,…,A_n ) § f(t⋅A_1,A_2,…,A_n )=t⋅f(A_1,A_2,…,A_n ) § f(A_1,A_2,…,A_i,..,A_j,…A_n )=−f(A_1,A_2,…,A_j,..,A_i,…A_n ) ○ Then f(A_1,…,A_n )=det(A_1,…,A_n )⋅f(I_1,…,I_n ) where § I_1=[1,0,0,…,0] § I_2=[0,1,0,…,0] § ⋮ § I_n=[0,0,0,…,1] • Proof ○ f(A_1,…,A_n ) ○ =f(a_11 I_1+a_12 I_2+…+a_1n I_n,…,a_n1 I_1+a_n2 I_2+…+a_nn I_n ) ○ =∑_█(1≤i_1,i_2,…,i_n≤n@all different)^n▒a_(1i_1 ) a_(2i_2 )…a_(ni_n )⋅f(I_(i_1 ),I_(i_2 ),…,I_(i_n ) ) ○ =∑_█(1≤i_1,i_2,…,i_n≤n@all different)^n▒a_(1i_1 ) a_(2i_2 )…a_(ni_n )⋅sign(i_1,…,i_n )⋅f(I_1,I_2,…,I_n ) ○ =f(I_1,I_2,…,I_n )⋅∑_█(1≤i_1,i_2,…,i_n≤n@all different)^n▒a_(1i_1 ) a_(2i_2 )…a_(ni_n )⋅sign(i_1,…,i_n ) ○ =f(I_1,I_2,…,I_n )⋅det(A_1,…,A_n ) • Example ○ |■8(A_(k×k)&0@C_(l×k)&B_(l×l) )|=|■(a_11&…&a_1k&&&@⋮&⋱&⋮&&&@a_k1&…&a_kk&&&@c_11&…&c_1k&b_11&…&b_1l@⋮&⋱&⋮&⋮&⋱&⋮@c_l1&…&c_lk&b_l1&…&b_ll )|=detA⋅detB ○ Consider a function f that satisfies the Uniqueness Theorem § f((A_1 ) ̅+(A_1 ) ̅ ̅,A_2,…,A_n )=f((A_1 ) ̅,A_2,…,A_n )+d((A_1 ) ̅ ̅,A_2,…,A_n ) § f(tA_1,A_2,…,A_n )=f(A_1,A_2,…,A_n ) § f(A_1,A_2,…,A_i,..,A_j,…A_n )=f(A_1,A_2,…,A_j,..,A_i,…A_n ) ○ Let f_BC (A_1,…,A_k )=|■8(A_(k×k)&0@C_(l×k)&B_(l×l) )| with B,C fixed, and A as variable § f_BC (A_1,…,A_k ) § =det(A_1,…,A_k ) f_BC (I_1,…,I_k ) § =detA⋅|■(1&&&&&@&⋱&&&&@&&1&&&@c_11&…&c_1k&b_11&…&b_1l@⋮&⋱&⋮&⋮&⋱&⋮@c_l1&…&c_lk&b_l1&…&b_ll )| § =detA⋅|■(1&&&&&@&⋱&&&&@&&1&&&@&&&b_11&…&b_1l@&&&⋮&⋱&⋮@&&&b_l1&…&b_ll )| § =detA⋅|■(I&@&B)| ○ Let g(B)=|■(I&@&B)| that satisfies the Uniqueness Theorem § g(B)=detB⋅g(I)=detB⋅|■(1&&@&⋱&@&&1)|=detB ○ Therefore |■8(A_(k×k)&0@0&B_(l×l) )|=detA⋅detB Properties of Determinant • det〖(AB)=detA⋅detB 〗 (where A_(n×n), B_(n×n)) ○ detA⋅detB ○ =|■8(A&0@I&B)| ○ =|■8(0&−AB@I&B)| ○ =(−1)^(n^2 ) |■8(I&B@0&−AB)| ○ =(−1)^(n^2 ) detI⋅det(−AB) ○ =(−1)^(n^2 )⋅det(−AB) ○ =(−1)^(n^2 ) (−1)^n det(AB) ○ =(−1)^(n^2+n) det(AB) ○ =det(AB) • Power of Determinants ○ det(A^n )=det(A⋅A…A)=det〖(A)⋅〗 det(A)…det(A)=(detA )^n • Determinant of Inverse ○ If A has an inverse(A^(−1)), and detA≠0, then ○ A^(−1) A=I ○ ⇒det〖A^(−1) 〗⋅detA=detI=1 ○ ⇒det〖A^(−1)=1/detA 〗 • Matrix Product and Determinant ○ |■8(A_(n×n)&0@I&B_(n×n) )| ○ =|■(a_11&…&a_1n&&&@⋮&⋱&⋮&&&@a_n1&…&a_nn&&&@1&&&b_11&…&b_1n@&⋱&&⋮&⋱&⋮@&&1&b_n1&…&b_nn )| ○ =|■(0&…&a_1n&−a_11 b_11&…&−a_11 b_1n@⋮&⋱&⋮&⋮&⋱&⋮@0&…&a_nn&−a_n1 b_11&…&−a_n1 b_1n@1&…&0&b_11&…&b_1n@⋮&⋱&⋮&⋮&⋱&⋮@0&…&1&b_n1&…&b_n )|=… ○ =|■(0&…&0&−∑_(i=1)^n▒〖a_1i b_i1 〗&…&−∑_(i=1)^n▒〖a_1i b_in 〗@⋮&⋱&⋮&⋮&⋱&⋮@0&…&0&−∑_(i=1)^n▒〖a_ni b_i1 〗&…&−∑_(i=1)^n▒〖a_ni b_in 〗@1&…&0&b_11&…&b_1n@⋮&⋱&⋮&⋮&⋱&⋮@0&…&1&b_n1&…&b_nn )| ○ =|■8(0&−AB@I&B)|](https://shawnzhong.com/wp-content/uploads/2017/11/img_5a0cf537a2c42.png)
