If you go to the zoo, the chimpanzee will wave to you. I do not understand their psyche, but I presume they do not understand what they are doing, except when they wave, they get food. I still remember seeing a tourist throw an ice-cream into the enclosure to the adorable chimp.
It’s sad when students are taught monkey tricks, i.e. techniques to solve problems without understanding. One example is at A level when they are required to diagonalize a 3×3 matrix (with distinct eigenvalues.) Some schools actually teach a method to find eigenvectors by using cross products. I know this because a few students asked me whether they can use that method in our course. I’m not sure how it goes, but I guess it’s this:
To find the basis for the nullspace of [tex]\lambda I – A[/tex], take two rows and compute the cross product. This works because the rowspace is the orthogonal complement of the nullspace and nullity=1.
The problem with this method is that it only works when A is 3×3 and the eigenspace is one dimensional, but I have a good feeling the students don’t know this. I recall a friend (JC teacher) who tested his students by asking them to diagonalize a 2×2 matrix and some of them who knew how to do it for 3×3 matrices couldn’t do it in this simpler case!