Math 108A – Assignment 5 Due: 05/10/21 20h00 PDT Instructions: Label your answers clearly

Math 108A – Assignment 5 Due: 05/10/21 20h00 PDT Instructions: Label your answers clearly. Create a PDF file, upload it to Gradescope, and instruct Gradescope on which pages to find your answers. In all questions you must provide a proof or justification for your answer. Your work will be graded for mathematical correctness and expository clarity. Please understand that, because of the size of the class and TA workload constraints, only selected problems may be graded. Point values will be determined at the time of grading. (1) Let V be a vector space over F, let T ∈ L(V ), and let W ⊂ V be a subspace invariant under T . Prove that null(T W ) = (null T ) ∩ W . (2) Let V be a vector space over F, let α ∈ F, and let T ∈ L(V ). Prove that T − αI is invertible if and only if α is not an eigenvalue of T . (3) Let V be a vector space over F and let T ∈ L(V ). Suppose that λ ∈ F is a eigenvalue of T with eigenvector v ∈ V . Prove that T n v = λn v, for all n ∈ N. (4) Find all eigenvalues and eigenvectors of the derivative map, D, on Pn (R). (5) Suppose that V and W are finite dimensional isomorphic vector spaces. Prove that L(V ) and L(W ) are isomorphic vector spaces. (6) Suppose that V is a 4-dimensional vector space over C and that {vj }4j=1 is a basis for V . Suppose that T ∈ L(V ) is an operator such that T v1 = 2v1 T v2 = 3v1 − iv2 T v3 = v1 + 2v2 − v3 T v4 = v3 + (1 + 2i)v4 . (a) Find the matrix of T relative to the basis {vj }4j=1 . (b) Find all eigenvalues of T . (c) Prove that T is invertible. (7) Let V be a finite dimensional vector space over F, and let T ∈ L(V ). Suppose that V has a basis of eigenvectors for T . Find the matrix of T relative to this basis. (8) Let V be a finite dimensional vector space over F, and let T ∈ L(V ). Let {λ1 , . . . , λm } be the distinct nonzero eigenvalues of T . Prove that dim E(λ1 , T ) + · · · + dim E(λm , T ) ≤ dim range T. Copyright © 2021 University of California