My view is that the people who get to matrix algebra didn't have problem understanding the random seeming operation of previous math courses. Consider the existence of the "FOIL" mnemonic for multiplying two binomials (https://en.wikipedia.org/wiki/FOIL_method ), or SOH-CAH-TOA, where it was easier for me to just remember the definitions than the acronym. I think those are hints that some people consider those to be 'random seeming'.
When I took algebra, I remember being tripped by all the new vocabulary. Why do mathematicians say something is "Abelian" instead of 'commutative'? Why is it important to have terms like "group", "ring", and "vector space"? Sure, I learned the definitions, but at that level they were 'random seeming'.
When I took algebra, I remember being tripped by all the new vocabulary. Why do mathematicians say something is "Abelian" instead of 'commutative'? Why is it important to have terms like "group", "ring", and "vector space"? Sure, I learned the definitions, but at that level they were 'random seeming'.