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seen Oct 23 at 21:14

Recognize that anything somebody can do, they will do. They'll throw exceptions, they'll assign objects to themselves, they'll use objects before giving them values, they'll give objects values and never use them, they'll give them huge values, they'll give them tiny values, they'll give them null values. In general, if it will compile, somebody will do it. As a result, make your classes easy to use correctly and hard to use incorrectly.

quadratic linear form XFX with matrix

F:  1   1
    1   1

in canonical base can be expressed as X' F' X'

X'  1   0   X'   
    0   0

where X'=A-1X, F'=AFA and A is transition matrix. Canonical base vectors are [1,0] and [-1,1]. Thus this is correct that (a+b)^2 = a^2 + b^2 + 2ab