Column space of a matrix

In this blog, i'm going to talk about column space of a matrix. 

 

1. column space : the column space is all the possible linear combinations of column vectors in a matrix.

It definitely contains the 0 vector, is closed under any scalar multiplication and addition. 

** Note that a matrix is just a way of writing a set of column vectors.

 

 2. Another way to view it :

If Ax is equal to that, and I'm saying that i can pick any vector x in Rn, I'm saying that I can pick all the possible values of the entries, all possible real values and all possible combinations of them, what is equal to?

you could say what are all of the possible vectors, or the set of all vectors you can create by taking linear combinations of the column vectors, or the span of column vectors, Or you could view it as, what are all of the possible values that Ax can take on if x is a member of Rn? 

** Note that if i try to set Ax to some value that it can't take on, clearly i'm not going to have any solution. If i'm able to find a solution, i'm able to find some x value.