Linear Algebra, 2020a

Section II. Geometry of Determinants 355
II
Geometry of Determinants
The prior section develops the determinant algebraically, by considering formulas
satisfying certain conditions. This section complements that with a geometric
approach. Beyond its intuitive appeal, an 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 the determinant. (The prior section
treats the determinant as a function of the rows but this section focuses on
columns.)
II.1
Determinants as Size Functions
This parallelogram picture is familiar from the construction of the sum of the
two vectors.
(
x2
y 2
)
(
x1
y 1
)
1.1 Definition In R n the box (or parallelepiped) formed by 〈⃗v 1 ,...,⃗v n 〉 is the
set {t 1 ⃗v 1 + ···+ t n ⃗v n | t 1 ,...,t n ∈ [0 ... 1]}.
Thus the parallelogram above is the box formed by 〈 ( x 1
y 1
)
,
( x2
y 2
)
〉. A three-space
box is shown in Example 1.4.
We can find the area of the above box by drawing an enclosing rectangle and
subtracting away areas not in the box.
B
A
y 2 D
y 1 C
F
E
x 2 x 1
area of parallelogram
= area of rectangle − area of A − area of B
− ···− area of F
=(x 1 + x 2 )(y 1 + y 2 )−x 2 y 1 − x 1 y 1 /2
− x 2 y 2 /2 − x 2 y 2 /2 − x 1 y 1 /2 − x 2 y 1
= x 1 y 2 − x 2 y 1
That the area equals the value of the determinant
∣ x 1 x ∣∣∣∣
2
= x
∣
1 y 2 − x 2 y 1
y 1 y 2