MAT3701 Assignment 4 2026 - Due 15 July 2026

MAT3701 Assignment 4 2026 - Due 15 July 2026 MAT3701 - Linear Algebra III ASSIGNMENT 04 Opens: 15 June 2026 Due: 15 July 2026 Instructions for the Assignment (1) Carefully explain all your arguments. (2) Only hand written PDF files will be accepted. This means handwritten on paper and then scanned. Solutions that look like they were generated by a computer will not be marked. (3) Late submissions will not be marked. E-mailed submissions will not be marked. No extensions will be granted. (4) Write your name, surname and student number on the first page. (5) Please attempt all the questions. Not all questions will be marked. Question 1 (14 marks) Let A =  1 1 1 1  .  1  is an eigenvector of A and determine its associated eigenvalue. (1.1) Verify that v1 = −1 [3]  1  is an eigenvector of A and determine its associated eigenvalue. (1.2) Verify that v2 = 1 [3] (1.3) Next, you will use the above column vectors to build a matrix Q s.t. Q −1 AQ is a diagonal matrix. (1.3.1) Write down the matrix Q. [2] (1.3.2) Write down Q −1 . [2] (1.3.3) Determine D if D = Q −1 AQ. For this step, the actual calculation will be marked, so please provide the details of the calculation. [4] □ Question 2 (10 marks) Let V be vector space over the field F , and let T : V → V be a linear operator. Suppose that λ ∈ F is an eigenvalue of T . Consider the set Eλ = {v ∈ V : T v = λv}. Prove that Eλ is a subspace of V . □ Question 3 (4 marks) Let  