Most of this will be completely incomprehensible if you aren’t mathematically trained.  I’m putting it out there anyway.

Thou shalt not confuse a vector space with its dual.  Thou shalt not add unnecessary structure without good cause.  Be careful not to identify objects related by noncanonical isomorphisms.  When composing isomorphisms, bear in mind the possibility of a “twist” when thou comest back to the same object again.

Do not favor subspaces over quotient spaces, nor quotient spaces over subspaces, but give equal justice to all methods of reduction.

Never be unjust to a mathematical entity, lest it rise up against you.  When something is taken away from a system, it is often needful to add something as well, to compensate it and to preserve its integrity.

Study a nonlinear thing by passing to a bigger linear algebra.

0 is a natural number.  Always include the identity, or the empty object, when you allow more than one of a thing.  Almost everything there can be two of, there can also be zero of.  But, if thou allowst only 1 of a thing, do not consider 0 to be a special case!  This rule of “0” holds in additive contexts; in multiplicative contexts “1” is nothing.

Know the properties of nothing.  Not by memorization, but in your soul.

“I came to wonder how much understanding anything (a term, a system, etc.) meant merely being used to it.  (Similarly I have observed with anything new in art, that when people say either that they cannot understand it or that they have come to understand it, that seems to mean largely either that they are not habituated to it or that they are).”  — St. T.S. Eliot.

You feel like you understand a thing, when you can represent it to your mind in multiple ways, and can also convert between them in your mind as needed.

Try to reverse premises and conclusions.  See if it still works.

You do not know a thing, until you know how it behaves in a greater and smaller number of dimensions.

real : complex : quaternionic :: orthogonal : linear : symplectic.