Linear Algebra, 2020a

Section III. Nilpotence 427
1.7 Example The transformation π: C 3 → C 3 projecting onto the first two coordinates
⎛
⎜
c ⎞ ⎛ ⎞
1 c 1
⎟
⎝c 2 ⎠ ↦−→
π ⎜ ⎟
⎝c 2 ⎠
c 3 0
has C 3 ⊃ R(π) =R(π 2 )=··· and {⃗0} ⊂ N (π) =N (π 2 )=··· where this is
the range space and the null space.
⎛
⎜
a
⎞
⎛
⎟
⎜
0
⎞
⎟
R(π) ={ ⎝b⎠ | a, b ∈ C} N (π) ={ ⎝0⎠ | c ∈ C}
0
c
1.8 Definition Let t be a transformation on an n-dimensional space. The generalized
range space (or closure of the range space) isR ∞ (t) =R(t n ). The
generalized null space (or closure of the null space) isN ∞ (t) =N (t n ).
This graph illustrates. The horizontal axis gives the power j of a transformation.
The vertical axis gives the dimension of the range space of t j as the
distance above zero, and thus also shows the dimension of the null space because
the two add to the dimension n of the domain.
n
nullity(t j )
dim(N ∞ (t))
0
rank(t j ) ...
dim(R ∞ (t))
0 1 2 j n
On iteration the rank falls and the nullity rises until there is some k such
that the map reaches a steady state R(t k )=R(t k+1 )=R ∞ (t) and N (t k )=
N (t k+1 )=N ∞ (t). This must happen by the n-th iterate.
Exercises
̌ 1.9 Give the chains of range spaces and null spaces for the zero and identity transformations.
̌ 1.10 For each map, give the chain of range spaces and the chain of null spaces, and
the generalized range space and the generalized null space.
(a) t 0 : P 2 → P 2 , a + bx + cx 2 ↦→ b + cx 2
(b) t 1 : R 2 → R 2 ,
( ( a 0
↦→
b)
a)