Visualizing equations in a row picture.

[NINHARDCOREFAN]

I don't understand how one visualizes in row pictures of equations. There is an example in the book:

With A=I(the identity matrix)
1x+0y+0z= 2
0x+1y+0z= 3
0x+0y+1z= 4

They drew these in the xyz plane. I don't know how they did this, can someone explain me that?

 

[HallsofIvy]

I don't even know what you mean by the "xyz plane"! I assume you meant an xyz coordinate system.
If you have trouble visualizing in 3 dimensions try reducing the problem to two first.
The line 1x+ 0y= 2 or x= 2 is a horizontal line of points (2, y) which is distance 2 above the x-axis. The line 0x+ 1y= 3 or y= 3 is a vertical line of points (x, 3) distance 3 to the right of the y-axis. They intersect at (2, 3).

In three dimensions, a single equation in x, y, z, represents a plane. The equation 1x+ 0y+ 0z= 2 or x= 2, corresponds to points (2, y, z) where y and z can be anything but x= 2. That's a plane parallel to the yz plane passing through (2, 0, 0). The equation 0x+1y+0z= 3 or x= 3 is the plane of points (x, 3, z) which is parallel to the xz plane and contains (0, 3, 0). The equation 0x+ 0y+ 1z= 4 or z= 4 is the plane of points (x, y, 4) which is parallel to the xy plane and distance 4 above it. Of course the three planes all intersect in the single point (2, 3, 4).