I'm learning linear algebra, specifically Gaussian Elimination. Please give me a short but detailed summary of based on the following chapter for my math notes!

5.4 Special Cases
So far, we have only encountered systems of equations which have exactly one solution. This is not always the case, as is demonstrated in
5x + 2y = 4, 10x + 4y = 13.
(5.20)
A normal attempt to solve this system would be to multiply the top equation by -2 and add to the second equation. This gives
{
J 5x + 2y = 4
0 = 5
(5.21)
Zero can never be equal to five, and yet we have followed the Gaussian elimination rules (Theorem 5.1) exactly. This is because this system of equations has no solution. It is easy to see why if we instead multiply the top equation by 2, i.e.,
[ 10x + 4y = 8,
10x + 4y = 13.
(5.22)
These two equations are simply contradicting each other. The left hand sides are exactly the same, and yet the right hand sides are different. Hence this system does not have a solution. Doing Gaussian elimination on such a system will result in contradictions of the type 0 = 5, which we encountered above. When this happens, it is safe to say that the system has no solution. By interpreting 5.10
the two equations in System of Equations (5.20) as two lines, it is possible to see what is going on graphically. The top equation, 5x + 2y = 4, represents the left line in Figure 5.10. The second equation, 10x+4y = 13, represent the rightmost line. As can be seen in the figure, these lines are parallel and will thus never cross. Since the crossing represents the solution, two parallel lines in the plane separated from each other is equivalent to a system of equations with no solution.
Having no solution is not the only special case. As an example, assume we have the following system of equations,
0.5
5x + 2y - 4 = 0
10x+4y-13 = 0
-x+ y =
8,
(5.23)
3x-3y=-24.

Performing Gaussian elimination on the first variable can be done by adding three times the first row to the second row. This gives
{
-x+y=8,
0 = 0.
3
(5.24)
The second row thus collapses to 0 = 0, which is always true, and no information about the y-variable is obtained. When this happens, there is not only one pair, (x, y), that satisfies the solution, but many. To express all such pairs, set y = t. This gives
and if x is solved for, we get
Ꮖ
Thus the vector
( )
can be written as
-x+ t = 8,
y=t,
x =
=-8+t, y= 0+t.
+
t. As we
5.11
have seen in Section 3.6.1, this is a line of the form P(t) = S+td. where S =
and d
(1)
In other words, there is an
entire line of solutions to this system of equations. This is shown in Illustration 5.11, where the green dot equals the point S, the red arrow shows the direction vector d, and the resulting line is shown in black.
Another way to look at this is to go back to the original equation,
-x+y= 8,
3x-3y24.
Now, if the bottom equation is multiplied by -, we get
-x+y=8,
-x+y=8.
9
8
(5.25)
(5.26)
10
6
P(t) = S+d
5
4
3
2
d
= (1)
(5.27)
-9 -8 -7 -6 -5 -4
-3
-2
-1
-1.
·8.
-2
O
(5.28)
Reset Forward

In other words, we get the same equation twice. The second equation can be seen as just repeating what we already knew, i.e., no new information is obtained. The two lines -x+y= 8 and 3x-3y=-24 are lying on top of each other, i.e., they are the same line. This is shown in Interactive Illustration 5.11 if you press the forward button below that illustration. In fact, it is the same line, only that −x + y = 8 and 3x-3y = -24 are written in implicit form, whereas P(t) = S+d is written in explicit form.
Now, consider a system with three equations and three unknowns, such as
2x+4y-2z = 16,
-x
-7y+2x= -27, 3y-6z=-21.
(5.29)
Just as in the case with the smaller two-equation systems, these systems can be interpreted geometrically. However, in this case every equation becomes a plane (see Section 3.6.2) in three-dimensional space. The solution to the system of equations is the point where all three planes intersect each other. This is shown in the first step of Interactive Illustration 5.12, where each plane represents one equation, and they all intersect in a single point, which is colored black in the figure. Note that you can change the viewpoint of the interactive illustration by holding the right mouse button down while moving the mouse (ctrl plus mouse button on mac, or pressing down with two fingers on mac laptop). On an iPad, change the camera position by dragging using two fingers.
5.12
Backward Reset || Forward
Interactive Illustration 5.12: One solution example: when all three planes intersect in one point, we have exactly one solution.
Sometimes however, the equations represent planes that all include the same line, as shown in the second step in Interactive Illustration 5.12. Then there is not only one solution, but infinitely many solutions to the system of equations. These solutions are equal to the line shown in black.

In other cases, two equations represent the same plane. This may be because they have the same coefficients, or one equation is equal to the other times a constant. Then there is an entire plane that satisfy these two equations, as illustrated by the dashed plane in the third step in Interactive Illustration 5.12. The solution to the entire system is then the intersection between this (double) plane and the third plane. This amounts to an entire line of solutions, as illustrated by the solid black line in the figure.
The last case when the system has a solution is when all three equations represent the same plane, such as the system
x+y+z = 1,
2x+2y+2z 2,
3x+3y+3z = 3.
(5.30)
If a point (x, y, z) satisfies the first equation, it also satisfies the second and third equations since they are just the first equation multiplied by 2 or 3. Therefore this point is also a solution to the entire system of equations. The solution is therefore visualised as entire black plane shown in the last step in Figure 5.12.
For a system of two equations and two unknowns not to have a solution, the two lines representing the two equations must be parallel as was shown in Interactive Illustration 5.10. This is not the case in three dimensions. Consider, for instance, the following system of equations.
3√6x + 3y+3z = √6 -3√6x+3y+3z = √6 -6y-6z= √6
(5.31)
It is easy to see that none of these planes are parallel, since their normals differ (and cannot be made equal by multiplying with a constant). The three normals are (3√√6, 3, 3), (-3√√6, 3, 3), and (0, -6, -6). Yet there exists no solution as is easily seen when performing Gaussian elimination. Adding the first and second row eliminates a from the first equation, i.e..
3√6x+3y+3z = √6, 6y+6z=2√6, -6y-6z= √6.
(5.32)
Now to eliminate y, we add the two last equations together, which results in
3√6x+3y+3z = √6, 6y+6z=2√√6, 0 = 3√6.
(5.33)
This also eliminates z and ends up in the contradiction 0 = 3√6, which means that no solutions exist. How this looks geometrically is shown in the first step in Figure 5.13. There are places where two of the equations are satisfied, but not the third, and these places are marked by dotted lines. Note that there is no point which is in all three planes, and thus no solution to the system of equations exists, although the planes are not parallel.

Backward
Reset Forward
Interactive Illustration 5.13: No solution example: three planes that intersect each other pairwise, but all three never intersect in a single point. The dashed lines are the intersections between two planes, and represent positions where two of the equations are satisfied but not the third.
If all three planes are parallel and non-coinciding, then there is no solution. This is shown in the second step of Figure 5.13.
A special case of this is when two equations represent the same plane, but the third equation represents a different plane parallel to, but not coinciding with, these. This example is shown in the third step of Figure 5.13, where the dashed plane satisfies two of the equations and the other plane satisfies the third equation. Again, no solution to this system of equations exists.
Finally, the last case when no solution exists is when two of the planes are parallel to each other (and different from each other). The third plane then intersects these two parallel planes along two lines (dashed in step 4 in Interactive Illustration 5.13), but there is not a single point that belongs to all three planes, and hence there exists no solution to the system of equations.
In summary, we can see that a system of equations either has % a system ... has
1. exactly one solution,
2. no solution, or
3. an infinite number of solutions.