Why is column space on the vertical in a matrix? Why is the "column space" on the vertical in a matrix? In my mind the column space is that space that the vectors in the matrix have created.
I mean, for example take the equations:
3x + 4y = 5

2x + 8y = 6

Then the matrix will be:
\begin{pmatrix}
3 & 4 \\
2 & 8 
\end{pmatrix}
But why is the space defined by this matrix on the vertical?
Aren't the two vectors:
3i + 4j

and
2i + 8j 

defining the space we're working in?
 A: There is also a space defined by the rows of the matrix and it is called (unsurprisingly) the "row space". When you construct a matrix from a linear system of equations, you are indeed constructing a matrix "row by row" and not "column by column" in the sense that each equation defines a row and not a column and so it might seem that you shouldn't care about the columns. However, when you begin solving the equation, you see that the columns also play an important role. For example, $Ax = b$ will have a solution if and only if $b$ belongs to the span of the columns of $A$ (the column space of $A$). Both spaces are important and are related by duality and/or the fact that you can always convert rows to columns by performing the transpose operation.
A: I think what you are meaning to ask is:

Why is the collumn space the range of the matrix, in the
  appropriate bases?

Otherwise, the collumn space is just a definition per se.
Therefore, we must see who is the image of the basis. That is, compute
$\begin{pmatrix}a_{1,1} & a_{1,2} & a_{1,3} &\cdots & a_{1,m} \\
a_{2,1} & a_{2,2} & a_{2,3} &\cdots & a_{2,m}  \\
\cdots & \cdots & \cdots &\cdots & \cdots \\
\cdots & \cdots & \cdots &\cdots & \cdots \\
a_{n,1} & a_{n,2} & a_{n,3} &\cdots & a_{n,m}  
\end{pmatrix} \cdot \begin{pmatrix}0 \\
\cdots \\
1 \\
\cdots \\
0
\end{pmatrix},$
where the $1$ appears on the $i-$th coordinate. But note that this is precisely
$\begin{pmatrix}a_{1,i} \\
a_{2,i} \\
\cdots \\
a_{(n-1),i} \\
a_{n,i}
\end{pmatrix}.$
