Linear Algebra

Thm: A matrix M is invertible <=> Null of M = 0 (trivial vector sapce)
So, if we can find a nonzero vector x, Mx=0, then M is not invertible.
Now, B is nxm (m>n), consider the system of linear eq. Bx=0, where x in R^m.
The linear system Bx=0 has n equations with m variables, then the dim.
of {x | Bx=0} >= m-n >0.
Thus, we can find a nonzero vector x such that Bx=0, and then ABx=0.
i.e. there exists a nonzero vector x, (AB)x=0, so, AB is not invertible.

2009-09-17 00:19:01 補充：
基本原理: Linear mapping 不可能將空間變大(增加dim.)
本題 B: R^m -> R^n 已經將空間 R^m變小了
再來A: R^n -> R^m 不可能 1對1.
OK!?

2009-09-17 00:53:12 補充：
物理系有這玩意兒嗎?

相後, 放在yahoo拍賣,
可能會有人喜歡收藏的,
儘管一試吧!

Rank(A) <= n
Rank(B) <= n

Rank(AB) <= n < m
Hence AB is not invertible

這是數學系的玩意