After reading it for a long time, it turned out to be two pictures, which are completely different from the theorem. In the theorem, r(2) is the rank of the vector group consisting of b 1, b2, b3 ..., while r in the following figure is the rank of the extended vector group, that is, (A 1, A2, A3...b), which is different here. Don't just think about a set of theorems, you don't need to remember theorems, and you can find different types of situations by learning deduction yourself. I won't give you proof of the theorem. As for why I didn't, I'll say it in general, because I took the postgraduate entrance examination on 20 12, and I haven't seen it for more than 10 months. First of all, I draw a conclusion from several rankings, no doubt.

r(a 1，a2....as)& lt； = r (a 1, a2...as, b), so why not? Because B can't be expressed linearly by a 1, A2 ... At this time, matrix/A 1, A2...As, B/ can't change the last column to 0, but if B can be expressed linearly by A 1, A2...As, then the last column can be changed to 0. The definition of linear transformation is combined here. Aortic second sound ... as is linear, which means equal to, unequal to, less than ... I'm sorry, but I don't know if you can understand it. Hehe, come on anyway. ....