Re: [線代] 線性系統

[Minkowski (四維之祖)]

※ 引述《mqazz1 (無法顯示)》之銘言：
: 1. If A is an m*n matrix and n<m, then the equation Ax=0
: will have infinite many answer.
:         False

         ┌ 1 ┐
　　A :＝│   │ is a 2 ×1 matrix, and Ax＝0 has a unique solution x＝0.
         └ 1 ┘


: =========
: 2. If the reduced row echelon form of [A b] contains a zero row,
: then Ax=b has infinitely many solutions
:         False

             ┌ 1 | 1 ┐
　　Let [A b] ＝ │   |   │. Then the reduced row echelon form of [A b]
             └ 1 | 1 ┘

   ┌ 1 | 1 ┐
is │   |   │, and Ax＝b has a unique solution x＝1.
   └ 0 | 0 ┘

: ===========
: 3. If (A1|b1) is obtained from (A|b) by a finite number of elementary row
: operations, then the systems A1x=b1 and Ax=b are equivalent
:         True

    Interchange two row                <=> Interchange two linear equation.

    Multiply a row by a nonzero number <=> Multiply a linear equation
                                           by a nonzeronumber.

    Multiply a row by a nonzero number <=> Multiply a linear equation
    and add it on another row              by a nonzero number and add it
                                           on another equation.
                                           ↑
                                          These action does not change
                                          the solution sets.

: ===========
: 4. The equation Ax=0 has the trivial solution if and only if
: thereare no free variables
:         False

(<=) Trivial.

         ┌ 1  0 ┐
(=>) B:＝│      │. Then Bx＝0 has the trivial solution and free variable.
         └ 0  0 ┘

　　But, the equation Ax=0 has only the trivial solution if and only if

there are no free variables.

: 請問這幾題是為什麼?
: 謝謝

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