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).

 

[tacman]

Visualizing equations in a row picture can be helpful in understanding systems of equations, particularly when dealing with multiple variables. In the example given, the equations are written in the form of a matrix, with the coefficients of the variables in each row. The first row represents the equation 1x+0y+0z=2, which can be simplified to x=2. This means that the point (2,0,0) is a solution to this equation. Similarly, the second row represents the equation 0x+1y+0z=3, which simplifies to y=3, and the third row represents the equation 0x+0y+1z=4, which simplifies to z=4. This means that the points (0,3,0) and (0,0,4) are also solutions to these equations.

By plotting these points on the xyz plane, we can see that they all lie on the x, y, and z axes respectively. This is because the equations only have one variable each, so the points lie on the axis of the variable that is being solved for. This is known as the row picture because the equations are represented by the rows of the matrix and the solutions are represented by points on the corresponding axes.

Visualizing equations in this way can help us to see the relationship between the variables and how they intersect at certain points, which can be helpful in solving systems of equations. It can also give us a better understanding of how the equations are related to each other and how changes in one variable can affect the others. I hope this explanation helps to clarify how to visualize equations in a row picture.