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mathematical_notation

The document introduces basic concepts of linear algebra including: 1) Common number sets such as real numbers, integers, and rational numbers. 2) Matrix operations including addition, scalar multiplication, and matrix multiplication. 3) Derivatives of functions with multiple variables are represented using partial derivatives. 4) Concepts of vectors, matrices, determinants, transpose and inner products are defined.

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⾔言 ⼊入⾨門     
•  ⽬目 •  •  ⼀一 •  •  •  •  •  ⾏行行
⽂文 ⽂文 ! ! !, " , # ! ! 2
•  N ⾃自 = {0, 1, 2, …} 0 •  Z = {…, -2, -1, 0, 1, 2, …} •  R •  C •  Q •  n ! N => n ⾃自 •  x ! Z => x
•  •  x A x A •  1 {1, 2, 3}, {3, 4, 5} { {1, 2}, {3, 4, 5}, {6, 7} } •  B A B A •  {1, 2} {1, 2, 3}, { {3, 4, 5} } { {1, 2}, {3, 4, 5}, {6, 7} } N!Z!Q!R!C •  (1) (2) (3) A B A B A B A B A B A B
•  ⼤大 2 •  1. S = {1, 3, 5, 7, 9} •  2. S = {x ! N | x % 2 = 1, x ! 10} S = {x | x ! N, x is odd and is less than or equal to 10} •  •  Haskell •  Python
•  •  a, b a x = b ! x = log a b •  ⾃自 e = 2.728! •   　 log e x log x ln x •  •  log(x ! y) = log x + log y •  log x n = n log x •  log(x / y) = log x ! log y •  e( x+y) = e x ! e y
d n •  (x n )! = (x ) = nx n"1 dx •  ( f ! g)" = f " ! g + f ! g" •  ( f (g(x)))! = f !(g(x))" g!(x) •  ⽅方 d "d % "d % 1 3 •  (log x 3 )! = $ logt ' ( $ x 3 ' = ( (3x 2 ) = dx # dt & # dx & t x t=x3 f(t)=log t, g(x)=x3 1 n+1 •  ! x n dx = x + C (n " #1) n= 1 ⺟母 0 n +1 !1 "x dx = log x + C
•  1 ⽤用 •  x = ( x1, x2 , x3, x4 ) f (x) = x 3 g(x) = x1 + x2 + x3 + x4 h(x) = (x, x 2 + x, x 3 ! x) •  •  ⽰示 •  f :R!R g : R4 ! R h : R ! R3
•  2 1 •  f (x, y, z) = x 3 + y 2 + z 5 !f !f (x, y, z) = f x (x, y, z) = 3x 2 (x, y, z) = f y (x, y, z) = 2y !x !y !f (x, y, z) = fz (x, y, z) = 5z 4 !z
⼠士 •  f ⼗十 = x, y f xy (x, y, z) = f yx (x, y, z) x y y x •  ⽤用 f xyzyyxwzxxzy (x, y, z) = f xxxxxyyyyzzzw (x, y, z)
•  1 •  f (x, y, z) = x 3 + y 2 + z 5 !f (x, y, z) = (3x 2 , 2y, 5z 4 ) ⾏行行 " % $ 3x 2 ' !f (x, y, z) = $ 2y ' $ ' # 5z 4 & •  /⾏行行 ⽤用 ⽂文 / •  (x, y, z) x !f (x) = (3x 2 , 2y, 5z 4 ) !x
•  2 ⾜足 "2 "2 "2 !f (x, y, z) = 2 f + 2 f + 2 f f : R3 ! R "x "y "z # "2 !f "2 "2 & =% 2 + 2 + 2 ( f !f $ "x "y "z ' = f xx + f yy + fzz . •  f (x, y, z) = x 3 + y 2 + z 5 !2 !2 !2 2 f = 6x f =2 f = 20z 3 !x !y 2 !z 2 "f (x, y, z) = 20z 3 + 6x + 2.
⾏行行
⾏行行 ⽅方 •  ⻑⾧長⽅方 •  ⾏行行 ⼤大⽂文 •  ⾏行行 ⼀一 •  n ⾏行行 m ⾏行行 n×m ⾏行行 ⾏行行 •  A = (aij ) i ⾏行行 j ⽬目 aij ⾏行行 column ! a12 ! a1m $ # a11 & ⾏行行 row # a21 a22 ! a1m & # & # " " # " & # an1 an2 ! anm & " %
⾏行行 •  A p×q ⾏行行 B q×r ⾏行行 •  A B ⾏行行 ⼀一 AB •  AB p×r ⾏行行 •  BA •  ⾏行行 ⽰示 q! •  A = (aij ), B = (bij ) AB i ⾏行行 j "a ik ! bkj k=1
⾏行行 ⾏行行 •  ⾏行行 (determinant) det A =| A | •  ⾏行行 ⽤用 •   　 ⽅方⾏行行 det ! a b $ •   　 　 　 　 　 　 A =# & det A = ad ! bc " c d % ! $ # a b c & •   　 　 　 　 　 　 　 A =# d e f & det A = aei + bfg + cdh ! afh ! bdi ! ceg # g h i & " % •  ⾏行行 n×n ⾏行行 A, B det AB = det A ! det B
⾏行行 •  transpose A t = AT = t A = T A •  •  ⾏行行 ⼊入 •  A i ⾏行行 j ⽬目 A j ⾏行行 i ⽬目 •  ! 1 $ # & •  A = (1 2 3) At = # 2 & # 3 & " % ! a b $ # & ! a c e $ •   　 　 　 　 　 　 A =# c d & At = # & # e f & # b d f & " % " %
⾏行行 •  ⾏行行 •  x, y n×1 , A n×n⾏行行 x t Ay •  n×n⾏行行 A 2 A  　 　 　 　 < x, y >  　 　 x!y < x, y >= x t Ay