Linear Algebra Lecture #2
[齋藤ｘｘ]は「齋藤正彦著『線型代数入門』東京大学出版会のｘｘページ参照」の意味。
[語句]
associative law 結合法則[齋藤 33,34]
augmented matrix 拡大係数行列[齋藤 54]
column exchange 列の交換(基本変形のひとつ) [齋藤 46]
commutative law 交換法則[齋藤 18,34]
digression 余談
dot product （ベクトルの）内積(= inner product)
elementary matrix 基本行列(elimination matrix とも言う) [齋藤 47]
elimination 消去(ここでは基本変形のこと) [齋藤 46]
Gauss ガウス(ドイツの数学者)
get a little leg up on the inverse 逆行列について少し先取りして学ぶ
go into the gory details of matrix multiplication 行列の積を非常に詳しく扱う
identity matrix 単位行列(unit matrix とも言う) [齋藤 35]
inverse matrix 逆行列[齋藤 41]
invertible 可逆な,逆行列を有する
matrix multiplication 行列の積[齋藤 33]
mention in passing ついでに触れる
parenthesis 括弧(複数形は parentheses)
permutation matrix 置換行列(行交換や列交換の行列) [齋藤 46]
pivot(掃き出すときの)かなめ[齋藤 48]
row exchange 行の交換
substitution 代入
subtract A from B B から A を引く
upper triangular matrix 上三角行列[齋藤 135]
I’m always sticking stuff in.
What is the matrix that does the whole elimination in one shot?
We’ re doing it on the fly.
outline of the lecture
1. Solve a system of linear equations not by determinant but by elimination.
When could elimination succeed and when could it fail?
2. back-substitution
3. elimination matrices
4. matrix multiplication (four ways)
solving by elimination
x + 2y + z = 2 (three equations with three unkowns)
3x + 8y + z = 12
4y + z = 2
elimination expressed in a matrix language
A x = b (A is a 3-by-3 matrix.)
first step of elimination
To eliminate x from equation 2, multiply equation 1 by 3 and subtract it from equation 2.
(The multiplier is 3.)
the first pivot ((1,1)-position)；the second pivot ((2,2)-position)
121
121
381
041
 0 2 －2
041
121
 0 2 －2 = U (upper triangular matrix)
005
The whole purpose of elimination is to get from A to U.
det A = det U = 10. (determinant)
The pivot is never zero. If the position is zero, exchange rows to get a non-zero pivot.
(If there is a zero pivot in U, the matrix is not invertible.)
Let me bring in the right hand side. (augmented matrix)
1212
121
2 121
2
3 8 1 12  0 2 –2 6 0 2 –2 6
= [U c]
0412
041
2 005
–10
back-substitution
x + 2y + z = 2
2y – 2z = 6
5 z = –10 (Solve the equations in the reverse order: z  y  x, be cause the matrix is upper triangular.)
Express the elimination steps in matrices. (elimination matrices)
step 1
???
121 121
? ? ? ×3 8 1  0 2 –2
???
041 041
???
3
? ? ? ×4 = 3×(column 1) + 4×(column 2) + 5×(column 3)
???
5
3×3 3×1 3×1
(matrix×column = column)
row operations
???
[1 2 7 ]
× ? ? ? = 1×(row 1) + 2×(row 2) + 7×(row 3) (row×matrix = row)
???
1×3
3×3 1×3
121
[1 0 0]×3 8 1  1 2 1
041
121
[–3 1 0]×3 8 1  0 2 –2
041
121
[0 0 1]×3 8 1  0 4 1
041
E2,1 (= elementary (or elimination) matrix that gets the (2,1)-position to be zero)
step 2
1 00
E3,2
= 0 10
0 –2 1
To put those steps together into a single matrix, (What matrix takes me from A to U?)
E3,2 (E2,1 A) = U
by the associative law, (E3,2 E2,1) A = U
permutation matrix
Exchange rows 1 and 2
0 1 0 (Exchange rows of the identity matrix.)
100
001
Exchange columns 1 and 2.
To do row operations, multiply on the left.
To do column operations, multiply on the right.
You can’t change the order of matrices. (The commutative law does not hold.)
(How to get from U back to A?)
the inverse matrix of E (here, the matrix that undoes the row operation)
E–1E = I (identity matrix) (E–1 “E to the minus one” or “E inverse”)