determinant axioms
determinant axioms
⎧⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎨⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎩{det(u,v)=det(u+sv,v)det(u,v)=det(u,v+su)translation invariance{det(su,v)=sdet(u,v)det(u,sv)=sdet(u,v)scaling{det(u1+u2,v)=det(u1,v)+det(u2,v)det(u,v1+v2)=det(u,v1)+det(u,v2)decomposition⇔⎧⎪
⎪
⎪⎨⎪
⎪
⎪⎩{det(u,v)=det(u+sv,v)det(u,v)=det(u,v+su)translation invariance{det(u1+su2,v)=det(u1,v)+sdet(u2,v)det(u,v1+sv2)=det(u,v1)+sdet(u,v2)linearity