Geometrical meaning of the Column Space Suppose I have $2$ planes in $R^3$ and they form a system $Ax=b$.    

I know the NullSpace of $A$ represents geometrically the vectors that form the intersection between the 2 planes shifted to the origin.   

I also know that the Row Space of A represents the span of the normal vectors to the 2 planes.     

But i was looking for some geometrical meaning for the Column Space of $A$.
They contain all the vectors $b$ that make $Ax=b$ have at least one solution, that i know.    

What I'm trying to see is what does the vectors from the Column Space mean in relating to the $2$ planes in $R^3$ or relating to the possible shiftings the system of planes could suffer to still yield a solution.
Well, they live in a separate, 'transpose' world. Now, as you have $2$ equations (of planes), $b$ and the columns will be elements of $\Bbb R^2$ and not $\Bbb R^3$.

I doubt they would represent anything special, related to the $2$ planes in $\Bbb R^3$, a strong reason is that if we multiply the equation of a plane (a row) by a scalar, we get the same plane, but the column space suffers a radical change..
Well, they live in a separate, 'transpose' world. Now, as you have $2$ equations (of planes), $b$ and the columns will be elements of $\Bbb R^2$ and not $\Bbb R^3$.

I doubt they would represent anything special, related to the $2$ planes in $\Bbb R^3$, a strong reason is that if we multiply the equation of a plane (a row) by a scalar, we get the same plane, but the column space suffers a radical change..