? asked in 科學數學 · 1 decade ago

線性代數的觀念題(nontrivial solutions是

線性代數的觀念題(nontrivial solutions是什麼)

A nonhomogeneous system has a nontrivial solution if and only if the numbers of equations of the system equal the numbers of the variables

這句話是錯的

為什麼?

Let A, B be m*n matrices

If both the system of linear equations Ax=0 and Bx=0 have nontrivial solutions

Then (A+B)x = 0 has nontrivial solution

這句話是錯的

為什麼?

還有請問一下

nontrivial solutions是什麼?

Update:

請問第二題的意思是

Ix = 0 ( I: identity matrix )

則x必為零解嗎?

3 Answers

Rating
  • 1 decade ago
    Favorite Answer

    各找一反例就成了:

    1. x+y=1; x+y=2 是個nonhomogeneous system having 2 equations for 2 variables. 然而它無解, 遑論a nontrivial solution.

    2. Let A=[2 0;1 2], B=[-1 0; -1 -1] be two 2 by 2 matrices having nontrivial solutions. Since A+B=[1 0; 0 1] = identity matrix, (A+B)x =0 has no nontrivial solutions.

    trivial solution: the solution with all variables equal to zero.

    nontrivial solution: a solution having at least one nonzero variable.

    有一位老師翻譯成無聊解與有聊解. 用不用隨便, 瞭解就好.

    2010-07-13 09:11:54 補充:

    Ix = 0 ( I: identity matrix )

    則x必為零解嗎?

    自然是囉. 只有當x的所有components 都是0時才會滿足Ix = 0. 或者這麼說 Ix = 0 恰有一解, the trivial solution.

  • 進哥
    Lv 7
    1 decade ago

    trivial 意指"不重要的,顯而易見的,瑣碎的,無聊的"

    trivial solution 一般都是指全為0的解,

    nontrivial solution 就是不全為0的解.

  • 興豪
    Lv 5
    1 decade ago

    nontrivial solutions 重要解答

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