this question is related to the elementary row operation on a matrix (multiply a row by a scalar, add the scalar multiple of one row to another, interchange two rows)
Q: Show that we can interchange any two rows in a matrix in 4 steps using the remaining two elementary row operations
regarding Orthogonal Matrices
Q:Find in terms of "n", the number of all n x n square orthogonal matrices with integers entries
regarding the 10 axioms that define a vector space
Q:Show , using the other 9 axioms, that in the definition of a vector space with regards to the 10 axioms, the axiom 3 (u + v = v + u [cumulative law]) is redundant.
I don't understand the problems in the manner they are presented. I would assume that I speak for the majority of common people when I say I work more efficiently with direct symbols rather than discussions on the parameters of how these symbols axiomatically operate (which in its self seems somewhat inheritly self-evident to me).
It's like the difference between learning English, and discussing each particle of a grammatical structure. The former is easier to learn intuitively rather than interpretively.
I'm a visual learner, so when dealing with mathematics, I always looked for visual patterns within the interactions of symbols. I have a harder time exploring thought via structured interpretation and descriptive break-down. I find that ambiguous descriptives tend to obfuscate the matter and muddle any potential to understand the subject as efficiently and fluently as one may desire. It starts to become so esoteric that you require a heavy background in whatever topic-area to even have the slightest clue what someone is attempting to describe. Descriptive teaching fails on so many different levels.
I am interested in how all of this works though. I took a browse through some of the material on algebraic matrixes and feel that yet again, there are intuitive patterns that are present that don't seem to be explored into enough.
Haven't you ever had that feeling when your teachers, from the moment you enter the school system to the time you follow school on your own terms, teach you a dozen different ways to do the same thing starting with what they assume is the easiest to pick up on?
If as children, we had all been exposed to the intricacies of mathematics from day one - we would all be masters of mathematics. We wouldn't need calculators, we could just automatically compute anything in our mind with outstanding accuracy and agility. The only problem is the way we teach things today. It's this horribly vague descriptive teaching method we use that prevents us from evolving our skills. It's a veil that holds us back from ever being enthusiastic about learning about higher mathematics for leisure.
If you show me a diagram representations of these questions, I think many people could answer them. But as it is, I might as well be using a shift cipher to imagine what you are trying to describe. Fight the system!