# 線性代數的觀念題(nontrivial solutions是

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

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

nontrivial solutions是什麼?

Update:

Ix = 0 ( I: identity matrix )

Rating

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.

• 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.

• nontrivial solutions 重要解答