Mistakes regarding proofs
I had a teacher who would repeat many times in different linear algebra classes "If you write 0 = 0 I'm going to give you a zero in the exam!". One of the most common operations is to multiply both sides of an equation by a number. But that number cannot be the zero because if we do that we are assuming that 0 = 0.
[math]\displaystyle{ a^x = a^y \iff x = y }[/math]. Suppose we do this [math]\displaystyle{ a^x - a^y = a^y - a^y \iff x - y = y - y }[/math]. Therefore, [math]\displaystyle{ x - y = 0 \iff x = y }[/math]. There is a mistake in this reasoning which is to assume that [math]\displaystyle{ 0^0 = 0 }[/math]. By definition there doesn't exist a number which raised to some power is equal to zero. Reciprocally, zero to the power zero is undefined. Another similar case happens if we try to "prove" that [math]\displaystyle{ 1 = 2 }[/math]. To do it we have to rely on some illegal division by 0.
Suppose we want to prove that [math]\displaystyle{ a + b = c + d }[/math]. We begin by saying that [math]\displaystyle{ a + b = x }[/math]. Then we say that [math]\displaystyle{ c + d = x }[/math]. Therefore [math]\displaystyle{ x = x \iff a + b = c + d }[/math]. Eureka! We didn't prove anything! What is wrong in the reasoning that we just did? The mistake is that we assumed the obvious to be true and vice-versa. If the statement is true, we want to prove that there isn't a case in which it's false. Else, if the statement is false, we want to prove that there isn't a case in which it's true. That's why in linear algebra and calculus many properties are true if we impose certain conditions. Other times we are presented with counter-examples to show that some property is true for some cases, but not for all of them.
Many proofs are done by finding a contradiction. Two statements that contradict each other because they can't be both false or both true at the same time. There is a very common pitfall here. Some statement implies another. Sometimes yes, other times no. For example: [math]\displaystyle{ a^2 + b^2 = c^2 }[/math]. We know that [math]\displaystyle{ a = b = c = 0 }[/math] is a solution and that [math]\displaystyle{ c^2 \geq 0 }[/math] because a sum of two squares can never be negative. However, we can't use the fact that [math]\displaystyle{ c^2 }[/math] is positive to state that there are infinite solutions to this equation. Suppose we say that a number is even. That doesn't imply that the number is positive or negative. Conversely, being negative or positive doesn't imply that it's even or not. For most purposes this logic is undeniable, we aren't concerned with the definition of what can be and what can't be denied. That discussion is way beyond what we learn at undergraduate levels.
A geometrical example of the reasoning above: we are given the points A, B and C and the statement that the distances between A and B is equal to A and C. There are two possibilities. Either A is between B and C and they form a straight line. Or A is not aligned to B and C and form an equilateral triangle. There is really no way to tell which case is it without having more information. The only way to prove this is to have more information and to rely on vectors.
With linear system of equations is specially common for people to end at absurd results such as 0 = 3 or the obvious 0 = 0. When we can't find variables to satisfy a linear system of equation it means that the system has no solution at all. Graphically, we have planes or lines that never intersect each other or maybe they do, but not all at the same point in space. Maybe they intersect in pairs for example. If we have an equation in a system that is a multiple of another, we can safely remove it from the system because it's not helping. It's the same as having a coordinate system of three linearly independent vectors in 3D space. If we try to build a coordinate system in 4D where the 4th coordinate is a linear combination of the previous three, it won't work at all! When we have an equation, isolate an unknown and then substitute in the same equation, we'll surely conclude that 0 = 0. In other words, we stated that a vector is parallel to itself or that a line intersects itself at every point.
