Linear Algebra

398 Chapter Five. Similarity⃗m ∈ M with t( ⃗m) ∉ M (for instance, the transformation that rotates the planeby a quarter turn does not map most members of the x = y line subspace backwithin that subspace). To ensure that the restriction of a transformation to apart of a space is a transformation on the part we need the next condition.2.7 Definition Let t: V → V be a transformation. A subspace M is t invariantif whenever ⃗m ∈ M then t( ⃗m) ∈ M (shorter: t(M) ⊆ M).Recall that Lemma III.1.4 shows that for any transformation t on an n dimensionalspace the rangespaces of iterates are stableas are the null spaces.R(t n ) = R(t n+1 ) = · · · = R ∞ (t)N (t n ) = N (t n+1 ) = · · · = N ∞ (t)Thus, the generalized null space N ∞ (t) and the generalized rangespace R ∞ (t)are t invariant. In particular, N ∞ (t − λ i ) and R ∞ (t − λ i ) are t − λ i invariant.The action of the transformation t − λ i on N ∞ (t − λ i ) is especially easy tounderstand. Observe that any transformation t is nilpotent on N ∞ (t), because if⃗v ∈ N ∞ (t) then by definition t n (⃗v) = ⃗0. Thus t − λ i is nilpotent on N ∞ (t − λ i ).We shall take three steps to prove this section’s major result. The next resultis the first.2.8 Lemma A subspace is t invariant if and only if it is t − λ invariant for allscalars λ. In particular, if λ i is an eigenvalue of a linear transformation t thenfor any other eigenvalue λ j the spaces N ∞ (t − λ i ) and R ∞ (t − λ i ) are t − λ jinvariant.Proof For the first sentence we check the two implications separately. The‘if’ half is easy: if the subspace is t − λ invariant for all scalars λ then usingλ = 0 shows that it is t invariant. For ‘only if’ suppose that the subspace is tinvariant, so that if ⃗m ∈ M then t( ⃗m) ∈ M, and let λ be a scalar. The subspaceM is closed under linear combinations and so if t( ⃗m) ∈ M then t( ⃗m) − λ ⃗m ∈ M.Thus if ⃗m ∈ M then (t − λ) ( ⃗m) ∈ M.The second sentence follows from the first. The two spaces are t−λ i invariantso they are t invariant. Apply the first sentence again to conclude that they arealso t − λ j invariant.QEDThe second step of the three that we will take to prove this section’s majorresult makes use of an additional property of N ∞ (t − λ i ) and R ∞ (t − λ i ), thatthey are complementary. Recall that if a space is the direct sum of two othersV = N ⊕ R then any vector ⃗v in the space breaks into two parts ⃗v = ⃗n + ⃗rwhere ⃗n ∈ N and ⃗r ∈ R, and recall also that if B N and B R are bases for N⌢and R then the concatenation B N BR is linearly independent (and so the twoparts of ⃗v do not “overlap”). The next result says that for any subspaces N and