- #1

Velo

- 17

- 0

$Ax=b$ is a system of linear equations with $m$ equations and $n$ variables. ${v}_{1}, {v}_{2}, ..., {v}_{n}$ are the vectors in the columns of $A$. The following are equivalent:

(1) The system $Ax=b$ is possible for every vector $b\in{\Bbb{R}}_{m}$.

(2) Every vector $b\in{\Bbb{R}}_{m}$ is a linear combination of $A$'s columns.

(3) $b\in span\left\{{v}_{1}, {v}_{2}, ..., {v}_{n}\right\}$ for every $b\in{\Bbb{R}}_{m}$.

(4) $span\left\{{v}_{1}, {v}_{2}, ..., {v}_{n}\right\}={\Bbb{R}}_{m}$

(5) $r(A)=m$

I don't get why $r(A)=m$ necessarily if the system is possible... Wouldn't, for example, the matrix:

1 1 1 | 0

0 0 1 | 0

0 0 0 | 0

Obtained after applying the Gauss theorem, be possible? Because x=-y, z = 0, and y could take on any arbitrary value?

And $r(A) = 2$, which is less than the initial number of equations...