Exercises 5A [[5A-1]] Suppose 𝑇 ∈ L(𝑉) and 𝑈 is a subspace of 𝑉. (a) Prove that if 𝑈 ⊆ null 𝑇,then 𝑈 is invariant under 𝑇. (b) Prove that if range 𝑇 ⊆ 𝑈,then 𝑈 is invariant under 𝑇. [[5A-2]] Suppose that 𝑇 ∈ L(𝑉) and 𝑉1, ..., 𝑉𝑚 are subspaces of 𝑉 invariant under 𝑇. Prove that 𝑉1 + ⋯ + 𝑉𝑚 is invariant under 𝑇. [[5A-3]] Suppose 𝑇 ∈ L(𝑉). Prove that the intersection of every collection of subspaces of 𝑉 invariant under 𝑇 is invariant under 𝑇. [[5A-4]] Prove or give a counterexample: If 𝑉 is finite-dimensional and 𝑈 is a sub- space of 𝑉 that is invariant under every operator on 𝑉, then 𝑈 = {0} or 𝑈 = 𝑉. [[5A-5]] Suppose 𝑇 ∈ L(𝐑2) is defined by 𝑇(𝑥, 𝑦) = (−3𝑦, 𝑥). Find the eigenvalues of 𝑇. [[5A-6]] Define 𝑇 ∈ L(𝐅2) by 𝑇(𝑤, 𝑧) = (𝑧, 𝑤). Find all eigenvalues and eigenvectors of 𝑇. [[5A-7]] Define 𝑇 ∈ L(𝐅3) by 𝑇(𝑧1, 𝑧2, 𝑧3) = (2𝑧2, 0, 5𝑧3). Find all eigenvalues and eigenvectors of 𝑇. [[5A-8]] Suppose 𝑃 ∈ L(𝑉) is such that 𝑃2 = 𝑃. Prove that if 𝜆 is an eigenvalue of 𝑃, then 𝜆 = 0 or 𝜆 = 1. [[5A-9]] Define 𝑇∶ 𝒫(𝐑) → 𝒫(𝐑) by 𝑇𝑝 = 𝑝′. Find all eigenvalues and eigenvectors of 𝑇. [[5A-10]] Define 𝑇 ∈ L(𝒫4(𝐑)) by (𝑇𝑝)(𝑥) = 𝑥𝑝′(𝑥) for all 𝑥 ∈ 𝐑. Find all eigenvalues and eigenvectors of 𝑇. [[5A-11]] Suppose 𝑉 is finite-dimensional, 𝑇 ∈ L(𝑉), and 𝛼 ∈ 𝐅. Prove that there exists 𝛿 > 0 such that 𝑇 − 𝜆𝐼 is invertible for all 𝜆 ∈ 𝐅 such that 0 < |𝛼 − 𝜆| < 𝛿. [[5A-12]] Suppose 𝑉 = 𝑈 ⊕ 𝑊, where 𝑈 and 𝑊 are nonzero subspaces of 𝑉. Define 𝑃 ∈ L(𝑉) by 𝑃(𝑢+𝑤) = 𝑢 for each 𝑢 ∈ 𝑈 and each 𝑤 ∈ 𝑊. Find all eigenvalues and eigenvectors of 𝑃. [[5A-13]] Suppose 𝑇 ∈ L(𝑉). Suppose 𝑆 ∈ L(𝑉) is invertible. (a) Prove that 𝑇 and 𝑆−1𝑇𝑆 have the same eigenvalues. (b) What is the relationship between the eigenvectors of 𝑇 and the eigenvec- tors of 𝑆−1𝑇𝑆? [[5A-14]] Give an example of an operator on 𝐑4 that has no (real) eigenvalues. [[5A-15]] Suppose 𝑉 is finite-dimensional, 𝑇 ∈ L(𝑉), and 𝜆 ∈ 𝐅. Show that 𝜆 is an eigenvalue of 𝑇 if and only if 𝜆 is an eigenvalue of the dual operator 𝑇′ ∈ L(𝑉′). [[5A-16]] Suppose 𝑣1,...,𝑣𝑛 is a basis of 𝑉 and 𝑇 ∈ L(𝑉). Prove that if 𝜆 is an eigenvalue of 𝑇, then |𝜆| ≤ 𝑛 max{∣M(𝑇)𝑗,𝑘∣ ∶ 1 ≤ 𝑗, 𝑘 ≤ 𝑛}, where M(𝑇)𝑗,𝑘 denotes the entry in row 𝑗, column 𝑘 of the matrix of 𝑇 with respect to the basis 𝑣1, ..., 𝑣𝑛. See Exercise 19 in Section 6A for a different bound on |𝜆|. [[5A-17]] Suppose𝐅=𝐑,𝑇∈L(𝑉),and𝜆∈𝐑.Provethat𝜆isaneigenvalueof𝑇 if and only if 𝜆 is an eigenvalue of the complexification 𝑇𝐂. See Exercise 33 in Section 3B for the definition of 𝑇𝐂. [[5A-18]] Suppose 𝐅=𝐑,𝑇 ∈ L(𝑉), and 𝜆 ∈ 𝐂. Prove that 𝜆 is an eigenvalue of the complexification 𝑇𝐂 if and only if 𝜆 is an eigenvalue of 𝑇𝐂. [[5A-19]] Show that the forward shift operator 𝑇 ∈ L(𝐅∞) defined by 𝑇(𝑧1,𝑧2,...) = (0,𝑧1,𝑧2,...) has no eigenvalues. [[5A-20]] Define the backward shift operator 𝑆 ∈ L(𝐅∞) by 𝑆(𝑧1,𝑧2,𝑧3,...) = (𝑧2,𝑧3,...). (a) Show that every element of 𝐅 is an eigenvalue of 𝑆. (b) Find all eigenvectors of 𝑆. Section 5A Invariant Subspaces ≈ 100√2 [[5A-21]] Suppose 𝑇 ∈ L(𝑉) is invertible. (a) Suppose 𝜆 ∈ 𝐅 with 𝜆 ≠ 0. Prove that 𝜆 is an eigenvalue of 𝑇 if and only if $\frac{1} {𝜆}$ is an eigenvalue of 𝑇−1. (b) Prove that 𝑇 and 𝑇−1 have the same eigenvectors. [[5A-22]] Suppose 𝑇 ∈ L(𝑉) and there exist nonzero vectors 𝑢 and 𝑤 in 𝑉 such that 𝑇𝑢=3𝑤 and 𝑇𝑤=3𝑢. Prove that 3 or −3 is an eigenvalue of 𝑇. [[5A-23]] Suppose 𝑉 is finite-dimensional and 𝑆, 𝑇 ∈ L(𝑉). Prove that 𝑆𝑇 and 𝑇𝑆 have the same eigenvalues. [[5A-24]] Suppose 𝐴 is an 𝑛-by-𝑛 matrix with entries in 𝐅. Define 𝑇 ∈ L(𝐅𝑛) by 𝑇𝑥 = 𝐴𝑥, where elements of 𝐅𝑛 are thought of as 𝑛-by-1 column vectors. (a) Supposethesumoftheentriesineachrowof𝐴equals1.Provethat1 is an eigenvalue of 𝑇. (b) Supposethesumoftheentriesineachcolumnof𝐴equals1.Provethat 1 is an eigenvalue of 𝑇. [[5A-25]] Suppose 𝑇 ∈ L(𝑉) and 𝑢, 𝑤 are eigenvectors of 𝑇 such that 𝑢 + 𝑤 is also an eigenvector of 𝑇. Prove that 𝑢 and 𝑤 are eigenvectors of 𝑇 corresponding to the same eigenvalue. [[5A-26]] Suppose 𝑇 ∈ L(𝑉) is such that every nonzero vector in 𝑉 is an eigenvector of 𝑇. Prove that 𝑇 is a scalar multiple of the identity operator. [[5A-27]] Suppose that 𝑉 is finite-dimensional and 𝑘 ∈ {1, ..., dim 𝑉 − 1}. Suppose 𝑇 ∈ L(𝑉) is such that every subspace of 𝑉 of dimension 𝑘 is invariant under 𝑇. Prove that 𝑇 is a scalar multiple of the identity operator. [[5A-28]] Suppose 𝑉 is finite-dimensional and 𝑇 ∈ L(𝑉). Prove that 𝑇 has at most 1 + dim range 𝑇 distinct eigenvalues. [[5A-29]] Suppose 𝑇 ∈ L(𝐑3) and −4, 5, and √7 are eigenvalues of 𝑇. Prove that thereexists𝑥∈𝐑3 such that 𝑇𝑥−9𝑥 = (−4,5,√7). [[5A-30]] Suppose 𝑇 ∈ L(𝑉) and (𝑇−2𝐼)(𝑇−3𝐼)(𝑇−4𝐼)=0. Suppose 𝜆 is an eigenvalue of 𝑇. Prove that 𝜆 = 2 or 𝜆 = 3 or 𝜆 = 4. [[5A-31]] Give an example of 𝑇 ∈ L(𝐑2) such that 𝑇4 = −𝐼. [[5A-32]] Suppose 𝑇 ∈ L(𝑉) has no eigenvalues and 𝑇4 = 𝐼. Prove that 𝑇2 = −𝐼. [[5A-33]] Suppose 𝑇 ∈ L(𝑉) and 𝑚 is a positive integer. (a) Prove that 𝑇 is injective if and only if 𝑇𝑚 is injective. (b) Prove that 𝑇 is surjective if and only if 𝑇𝑚 is surjective. [[5A-34]] Suppose 𝑉 is finite-dimensional and 𝑣1, ..., 𝑣𝑚 ∈ 𝑉. Prove that the list 𝑣1, ..., 𝑣𝑚 is linearly independent if and only if there exists 𝑇 ∈ L(𝑉) such that 𝑣1, ..., 𝑣𝑚 are eigenvectors of 𝑇 corresponding to distinct eigenvalues. [[5A-35]] Suppose that 𝜆1, ..., 𝜆𝑛 is a list of distinct real numbers. Prove that the list 𝑒𝜆1𝑥, ..., 𝑒𝜆𝑛𝑥 is linearly independent in the vector space of real-valued functions on 𝐑. Hint: Let 𝑉 = span(𝑒𝜆1𝑥,...,𝑒𝜆𝑛𝑥), and define an operator 𝐷 ∈ L(𝑉) by 𝐷 𝑓 = 𝑓 ′. Find eigenvalues and eigenvectors of 𝐷. [[5A-36]] Suppose that 𝜆1, ..., 𝜆𝑛 is a list of distinct positive numbers. Prove that the list cos(𝜆1𝑥), ..., cos(𝜆𝑛𝑥) is linearly independent in the vector space of real-valued functions on 𝐑. [[5A-37]] Suppose 𝑉 is finite-dimensional and 𝑇 ∈ L(𝑉). Define 𝒜 ∈ L(L(𝑉)) by 𝒜(𝑆) = 𝑇𝑆 for each 𝑆 ∈ L(𝑉). Prove that the set of eigenvalues of 𝑇 equals the set of eigenvalues of 𝒜. [[5A-38]] Suppose 𝑉 is finite-dimensional, 𝑇 ∈ L(𝑉), and 𝑈 is a subspace of 𝑉 invariant under 𝑇. The quotient operator 𝑇/𝑈 ∈ L(𝑉/𝑈) is defined by (𝑇/𝑈)(𝑣+𝑈) = 𝑇𝑣+𝑈 for each 𝑣 ∈ 𝑉. (a) Show that the definition of 𝑇/𝑈 makes sense (which requires using the condition that 𝑈 is invariant under 𝑇) and show that 𝑇/𝑈 is an operator on 𝑉/𝑈. (b) Show that each eigenvalue of 𝑇/𝑈 is an eigenvalue of 𝑇. [[5A-39]] Suppose 𝑉 is finite-dimensional and 𝑇 ∈ L(𝑉). Prove that 𝑇 has an eigenvalue if and only if there exists a subspace of 𝑉 of dimension dim 𝑉 − 1 that is invariant under 𝑇. [[5A-40]] Suppose 𝑆, 𝑇 ∈ L(𝑉) and 𝑆 is invertible. Suppose 𝑝 ∈ 𝒫(𝐅) is a polynomial. Prove that 𝑝(𝑆𝑇𝑆−1) = 𝑆𝑝(𝑇)𝑆−1. [[5A-41]] Suppose 𝑇 ∈ L(𝑉) and 𝑈 is a subspace of 𝑉 invariant under 𝑇. Prove that 𝑈 is invariant under 𝑝(𝑇) for every polynomial 𝑝 ∈ 𝒫(𝐅). [[5A-42]] Define 𝑇 ∈ L(𝐅𝑛) by 𝑇(𝑥1, 𝑥2, 𝑥3, ..., 𝑥𝑛) = (𝑥1, 2𝑥2, 3𝑥3, ..., 𝑛𝑥𝑛). (a) Find all eigenvalues and eigenvectors of 𝑇. (b) Find all subspaces of 𝐅𝑛 that are invariant under 𝑇. [[5A-43]] Suppose that 𝑉 is finite-dimensional, dim 𝑉 > 1, and 𝑇 ∈ L(𝑉). Prove that {𝑝(𝑇) ∶ 𝑝 ∈ 𝒫(𝐅)} ≠ L(𝑉).