Linear Algebra, 2020a

Section III. Nilpotence 425
has this second power (
a
c
and this third power. (
a
c
) ( )
b t
↦−→
2 a b
d 0 0
) ( )
b t
↦−→
3 b a
d 0 0
After that, t 4 = t 2 and t 5 = t 3 , etc.
1.3 Example Consider the shift transformation t: C 3 → C 3 .
⎛ ⎞ ⎛ ⎞
x 0
⎜ ⎟
⎝y⎠
↦−→
t ⎜ ⎟
⎝x⎠
z y
We have that
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛
x 0 0
⎜ ⎟
⎝y⎠
↦−→
t ⎜ ⎟
⎝x⎠
↦−→
t ⎜ ⎟
⎝0⎠
t ⎜
0
⎞
⎟
↦−→ ⎝0⎠
z y x 0
so the range spaces descend to the trivial subspace.
⎛ ⎞
⎛
0
⎜ ⎟
R(t) ={ ⎝a⎠ | a, b ∈ C} R(t 2 ⎜
0
⎞
⎛
⎟
)={ ⎝0⎠ | c ∈ C} R(t 3 ⎜
0
⎞
⎟
)={ ⎝0⎠}
b
c
0
These examples suggest that after some number of iterations the map settles
down.
1.4 Lemma For any transformation t: V → V, the range spaces of the powers
form a descending chain
V ⊇ R(t) ⊇ R(t 2 ) ⊇···
and the null spaces form an ascending chain.
{⃗0} ⊆ N (t) ⊆ N (t 2 ) ⊆···
Further, there is a k>0such that for powers less than k the subsets are
proper: if if j