Linear Algebra

Section II. Geometry of Determinants 321
1.2 Remark Although property (2) of the definition of determinants is redundant,
it raises an important point. Consider these two.
�v
�u
� �
�
�4
1�
�
�2 3�
=10
� �
�
�1
4�
�
�3 2�
= −10
The only difference between them is in the order in which the vectors are taken.
If we take �u first and then go to �v, follow the counterclockwise arc shown, then
the sign is positive. Following a clockwise arc gives a negative sign. The sign
returned by the size function reflects the ‘orientation’ or ‘sense’ of the box. (We
see the same thing if we picture the effect of scalar multiplication by a negative
scalar.)
Although it is both interesting and important, the idea of orientation turns
out to be tricky. It is not needed for the development below, and so we will pass
it by. (See Exercise 27.)
1.3 Definition The box (or parallelepiped) formed by 〈�v1,...,�vn〉 (where each
vector is from Rn �
) includes all of the set {t1�v1 + ···+ tn�vn � t1,... ,tn ∈ [0..1]}.
The volume of a box is the absolute value of the determinant of the matrix with
those vectors as columns.
1.4 Example Volume, because it is an absolute value, does not depend on
the order in which the vectors are given. The volume of the parallelepiped in
Exercise 1.1, can also be computed as the absolute value of this determinant.
� �
�
�0
2 0�
�
�
�3
0 3�
� = −12
�1
2 1�
The definition of volume gives a geometric interpretation to something in
the space, boxes made from vectors. The next result relates the geometry to
the functions that operate on spaces.
1.5 Theorem A transformation t: R n → R n changes the size of all boxes by
the same factor, namely the size of the image of a box |t(S)| is |T | times the
size of the box |S|, where T is the matrix representing t with respect to the
standard basis. That is, for all n×n matrices, the determinant of a product is
the product of the determinants |TS| = |T |·|S|.
The two sentences state the same idea, first in map terms and then in matrix
terms. Although we tend to prefer a map point of view, the second sentence,
the matrix version, is more convienent for the proof and is also the way that
we shall use this result later. (Alternate proofs are given as Exercise 23 and
Exercise 28.)
�v
�u