Geometric intuition behind linear system Ax=B? between ways of view?

Question

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.

Answer 1

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.