Re: [線代] 線性系統
※ 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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08/26 21:16, , 1F
08/26 21:16, 1F
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