Linear Algebra

374 Chapter Five. Similarityand this third power. (ac) ( )b t↦−→3 b ad 0 0After 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 yWe have that⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛x 0 0⎜ ⎟⎝y⎠↦−→t ⎜ ⎟⎝x⎠↦−→t ⎜ ⎟⎝0⎠t ⎜0⎞⎟↦−→ ⎝0⎠z y x 0so the range spaces descend to the trivial subspace.⎛ ⎞⎛ ⎞⎛0⎜ ⎟R(t) = { ⎝a⎠ ∣ 0a, b ∈ C} R(t 2 ⎜ ⎟) = { ⎝0⎠ ∣ c ∈ C} R(t 3 ⎜0⎞⎟) = { ⎝0⎠}bc0These examples suggest that after some number of iterations the map settlesdown.1.4 Lemma For any transformation t: V → V, the range spaces of the powersform a descending chainV ⊇ R(t) ⊇ R(t 2 ) ⊇ · · ·and the null spaces form an ascending chain.{⃗0} ⊆ N (t) ⊆ N (t 2 ) ⊆ · · ·Further, there is a k such that for powers less than k the subsets are proper so thatif j < k then R(t j ) ⊃ R(t j+1 ) and N (t j ) ⊂ N (t j+1 ) while for higher powers thesets are equal, that is, if j k then R(t j ) = R(t j+1 ) and N (t j ) = N (t j+1 )).Proof First recall that for any map the dimension of its range space plusthe dimension of its null space equals the dimension of its domain, So if thedimensions of the range spaces shrink then the dimensions of the null spaces mustgrow. We will do the range space half here and leave the rest for Exercise 14.We start by showing that the range spaces form a chain. If ⃗w ∈ R(t j+1 ), sothat ⃗w = t j+1 (⃗v), then ⃗w = t j ( t(⃗v) ). Thus ⃗w ∈ R(t j ).Next we verify the “further” property: while the subsets in the chain ofrange spaces may be proper for a while, from some power k onward the rangespaces are equal. We first show that if any pair of adjacent range spaces inthe chain are equal R(t k ) = R(t k+1 ) then all subsequent ones are also equal