"Normalize" a vector usually refers to transforming it into a vector that points in the same direction, but has unit length.
Det stands for determinant, which is easy to define for 2x2 and 3x3 matrices. It has nice geometrical properties.
... you may want to narrow your question about linear algebra...
It is basically the study of transformations whose result scale according to what happens to their input (using twice the original entry gives twice the original answer, and if you have two entries, the same answer results from (a) adding first and then transforming, and (b) transforming each and then adding).
That's the starting point, but lots of things can be said about such transformations.
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