Re: [線代] row echelon form
※ 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
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