Geometric intuition behind linear system Ax=B? between ways of view? I'm trying to pool how people see a simple Ax=B system from
a geometric point of view.
So far, I can regard it, 
as a set of equations, that can be viewed as hyperplanes,
or as linear combinations of column vectors scaled by x components that would reach B, or as seeing B as coordinates from identity to X, this one in local coordinates to transformation axes A.
I'm always asking myself if, those ideas are related between them by geometric means; I mean, if you could use a compass, rulers, protractor, draw unit circles, etc, transfer vector components to others dimensions or so, we would reach to a unified view of those?
I'm very dull at maths, thanks for the help.
 A: well i managed to have some kind of view to a simple A (2x2) x (2x1) = B (2x1), there are for sure many other possibilities.
Assuming that the hyperplane equations are normalized, A, b to n/||n|| . x = ||n||
I used 2 cartesian planes rotated by 90º in y axis, around the unit center, then:


*

*draw the same column vectors of A (A:1,A:2) and B on the two planes

*draw also the A vectors (A:1 x1,A:2 x2) scaled until the sum reaches B

*transfer the x coordinates of both vectors to the center making the first transpose vector of A (A1:,A2:); 

*rotate 90º in (cartesian planes local) z, the y coordinates of both column vectors of A and B and transfer to the center, making the second transpose vector of B;

*this will result on the row vectors of A and quantities of B laying on the z plane

*use the transfered B quantities to make two circles

*scale the transpose vectors of A to the B circles, both already on a z plane with that common center (they should be the true normals to the line equations, since A rows are unit vectors)

*draw the ortogonal lines passing by those last, and make the proper intersection

*assuming that the cartesian planes are on z = 1, draw a ray one unit behind each cartesian plane on their z axis, passing to each of the column vectors; for both planes, bring the planes to, aligning it to the scaled A column vectors; the quantity laying on the z axis of the planes would be the X scales (analog to homogeneous coordinates)

*now transfer those scales laying in both z-axis (local to the each cartesian plane) by 1 and 1-x1, to the center, they should align equally to the x, z coordinates of the intersection.


the symmetry isnt good, because i had to rotate the 2nd set of y-coordinates to get the transposes. Also in the last push, i had to correct the last push to 1-x1. but this is the best i could get.
the whole cenario looks like a cdrom: 





sorry if my question wasn't clear enough, i wanted a geometric setting for a simple Ax=B showing those two views, just using rotations and translations.
