Linear Algebra

Section III. Nilpotence 375R(t k+1 ) = R(t k+2 ), etc. This holds because t: R(t k+1 ) → R(t k+2 ) is the samemap, with the same domain, as t: R(t k ) → R(t k+1 ) and it therefore has thesame range R(t k+1 ) = R(t k+2 ) (it holds for all higher powers by induction). Soif the chain of range spaces ever stops strictly decreasing then from that pointonward it is stable.We end by showing that the chain must eventually stop decreasing. Eachrange space is a subspace of the one before it. For it to be a proper subspace itmust be of strictly lower dimension (see Exercise 12). These spaces are finitedimensionaland so the chain can fall for only finitely-many steps, that is, thepower k is at most the dimension of V.QED1.5 Example The derivative map a + bx + cx 2 + dx 3 d/dx↦−→ b + 2cx + 3dx 2 on P 3has this chain of range spacesR(t 0 ) = P 3 ⊃ R(t 1 ) = P 2 ⊃ R(t 2 ) = P 1 ⊃ R(t 3 ) = P 0 ⊃ R(t 4 ) = {⃗0}(all later elements of the chain are the trivial space). And it has this chain ofnull spacesN (t 0 ) = {⃗0} ⊂ N (t 1 ) = P 0 ⊂ N (t 2 ) = P 1 ⊂ N (t 3 ) = P 2 ⊂ N (t 4 ) = P 3(later elements are the entire space).1.6 Example Let t: P 2 → P 2 be the map c 0 + c 1 x + c 2 x 2 ↦→ 2c 0 + c 2 x. As thelemma describes, on iteration the range space shrinksR(t 0 ) = P 2 R(t) = {a + bx ∣ ∣ a, b ∈ C} R(t 2 ) = {a ∣ ∣ a ∈ C}and then stabilizes R(t 2 ) = R(t 3 ) = · · · while the null space growsN (t 0 ) = {0} N (t) = {cx ∣ ∣ c ∈ C} N (t 2 ) = {cx + d ∣ ∣ c, d ∈ C}and then stabilizes N (t 2 ) = N (t 3 ) = · · · .1.7 Example The transformation π: C 3 → C 3 projecting onto the first two coordinates⎛⎜c ⎞ ⎛1⎟⎝c 2 ⎠π ⎜c ⎞1⎟↦−→ ⎝c 2 ⎠c 3 0has C 3 ⊃ R(π) = R(π 2 ) = · · · and {⃗0} ⊂ N (π) = N (π 2 ) = · · · where this isthe range space and the null space.⎛⎜a⎞⎛⎟R(π) = { ⎝b⎠ ∣ ⎜0⎞⎟a, b ∈ C} N (π) = { ⎝0⎠ ∣ c ∈ C}0c1.8 Definition Let t be a transformation on an n-dimensional space. The generalizedrange space (or the closure of the range space) is R ∞ (t) = R(t n ) Thegeneralized null space (or the closure of the null space) is N ∞ (t) = N (t n ).