Re: [線代] row echelon form

[Minkowski (四維之祖)]

※ 引述《mqazz1 (無法顯示)》之銘言：
: (a) If (A|B) can be obtained from (C|D) by a finite sequence of elementary
: column operations, then the systems Ax=B and Cx=D are equivalent
:         False

                      ┌ 1 ┐      ┌ 0 ┐
　　Set A ＝ B ＝ C ＝│   │, D ＝│   │. Then
                      └ 0 ┘      └ 0 ┘

         ┌ 1  1 ┐                                  ┌ 1  0 ┐
(A|B) ＝ │      │ is column equivalent to (C|D) ＝ │      │. But
         └ 0  0 ┘                                  └ 0  0 ┘

Ax＝B has solution x ＝ 1, and Cx＝D has solution x ＝ 0.

: =========
: (d) If (A|B) is in row echelon form, then the system AX=B must have a solution
:         False

            ┌ 0 ┐      ┌ 1 ┐               ┌ 0  1 ┐
　　Let A ＝│   │, B ＝│   │. Then (A|B) ＝│      │ is in row echelon
            └ 0 ┘      └ 0 ┘               └ 0  0 ┘

form. But Ax＝B has no solution.

: ============
: (c) If a system of linear equations has two different solutions,
: then it must have infinitely many solutions
:         False

 Warning!   One may get different answer in this problem over different

field! The answer is trivial when both the number of equations and the

order of the field you discussing are finite, FALSE. But, if these system

is consider over an infinite field, it is TRUE.

[Infinite field case]

　　Let x≠y and Ax ＝ Ay ＝ b. Then A(x-y) ＝ 0 for nonzero vector x-y.

Let z ＝ x-y. Then for any scalar λ, A(x＋λz) ＝ Ax ＋ λAz ＝ b.

There are infinitely many vectors in the form x＋λz.

: 請問這些是為什麼?
: 謝謝

--
※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 140.123.62.134
推 mqazz1      :謝謝 請問如果(d)改成
→ mqazz1      :(A|B) is in reduce row echelon form會是true嗎?
→ Minkowski   :我給的那個例子剛好就是