Linear Algebra

Section II. Geometry of Determinants 319
4.II Geometry of Determinants
The prior section develops the determinant algebraically, by considering what
formulas satisfy certain properties. This section complements that with a geometric
approach. One advantage of this approach is that, while we have so far
only considered whether or not a determinant is zero, here we shall give a meaning
to the value of that determinant. (The prior section handles determinants
as functions of the rows, but in this section columns are more convenient. The
final result of the prior section says that we can make the switch.)
4.II.1 Determinants as Size Functions
This parallelogram picture
� �
x2
y2
� �
x1
is familiar from the construction of the sum of the two vectors. One way to
compute the area that it encloses is to draw this rectangle and subtract the
area of each subregion.
y 2
y 1
A
C
x 2
B
E
x 1
D
F
y1
area of parallelogram
= area of rectangle − area of A − area of B
−···−area of F
=(x1 + x2)(y1 + y2) − x2y1 − x1y1/2
− x2y2/2 − x2y2/2 − x1y1/2 − x2y1
= x1y2 − x2y1
The fact that the area equals the value of the determinant
� �
� �
� �
� = x1y2 − x2y1
� x1 x2
y1 y2
is no coincidence. The properties in the definition of determinants make reasonable
postulates for a function that measures the size of the region enclosed
by the vectors in the matrix.
For instance, this shows the effect of multiplying one of the box-defining
vectors by a scalar (the scalar used is k =1.4).
�w
�v
�w
k�v