Linear Algebra

Section II. Geometry of Determinants 321⃗w⃗w⃗vk⃗v(On the right the rescaled region is in solid lines with the original region inshade for comparison.) That is, we can reasonably expect of a size measure thatsize(. . . , k⃗v, . . . ) = k · size(. . . ,⃗v, . . . ). Of course, this property is familiar fromthe definition of determinants.Another property of determinants that should apply to any function givingthe size of a box is that it is unaffected by combining rows. Here are beforecombiningand after-combining boxes (the scalar shown is k = 0.35). The boxformed by v and k⃗v + ⃗w is more slanted than the original one but the two havethe same base and the same height and hence the same area.⃗wk⃗v + ⃗w⃗v⃗v(As before, the figure on the right has the original region in shade for comparison.)So we expect that size(. . . ,⃗v, . . . , ⃗w, . . . ) = size(. . . ,⃗v, . . . , k⃗v + ⃗w, . . . ); again, arestatement of a determinant postulate.Lastly, we expect that size(⃗e 1 ,⃗e 2 ) = 1⃗e 2and we naturally extend that to any number of dimensions size(⃗e 1 , . . . ,⃗e n ) = 1.Because property (2) of determinants is redundant (as remarked followingthe definition) we have that the properties of the determinant function arereasonable to expect of a function that gives the size of boxes. The prior sectionstarts with these properties and shows that the determinant exists and is unique,so we know that these postulates are consistent and that we do not need anymore postulates. Thus, we will interpret det(⃗v 1 , . . . ,⃗v n ) as the size of the boxformed by the vectors.1.2 Remark Although property (2) of the definition of determinants is redundantit raises an important point. Consider these two.⃗e 1⃗v⃗v⃗u⃗u∣ 4 1∣ ∣∣∣ 2 3∣ = 10 1 43 2∣ = −10Swapping changes the sign. On the left we take ⃗u first in the matrix and thenfollow the counterclockwise arc to ⃗v, following the counterclockwise arc, and geta positive size. On the right following the clockwise arc gives a negative size.