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Main Question or Discussion Point
How do we show that, given a matrix $A$, the sign of the determinant is positive or negative depending on the orientation of the rows of A, with respect to the standard orientation of $R^n$?
If you refer to the attached file, this matix has a positive orientation (and the sign of the determinant is positive) since the direction from (a,b) to (c,d) is counterclockwise, which is the same orientation as R2 (counterclockwise). Thanks!Isn't that the very definition of "the rows making up the columns of A have the same (res. the opposite) orientation as the standard basis" ?
If not, write the definition of orientation. you're using.