Ax=b; every vector b is exactly from one vector x (from row space of A) <more>?

[kthurst]

"Ax=b; every vector b is exactly from one vector x (from row space of A)".. <more>?

Hi,
I m referring 'Linear Algebra and its applications by Gilbert Starng".

I read (ch.3.1)"Matrix transforms every vector from its row space to its column space". Or
if given Ax=b; every vector b is exactly from one vector x (from row space of A).
Just want to know What if we multiply A with some vector x which is not in row space?
(can we do it !?)...
Not able to figure it out. may be i have missed some basic concept or misunderstood it.

Please help me out..
than u..

 

[fresh_42]

$A\, : \,V\longrightarrow V$ defines a linear map on the vector space $V$. We can write $V=\operatorname{im} A \oplus \ker A$. This means we can split every vector $v=v_{i}+ v_k$ where $v_i$ is in the row space and $v_k$ is sent to zero.