HW1.pdf - Introduction to Optimization MS&E 111/MS&E...

This preview shows page 1 - 2 out of 3 pages.

Introduction to Optimization MS&E 111/MS&E 211/ENGR 62 HW1 Course Instructor: Ashish Goel Due Date: April 15, 2021, 1:30pm PDT Please refer to the syllabus for submission instructions, in particular those pertaining to Excel work. Having taken the relevant screenshots, please add them as part of the rest of the write-up in a single pdf file. Please be sure to assign pages in your solution to the relevant problems on Gradescope. The teaching staff will not be responsible for grading unassigned pages. Problem 1 (20 Points) Please answer each of the following questions that would serve as a Linear Algebra review: 1. In summation notation, the i, j entry of AB is ( AB ) ij = X k a ik b kj . If A and B are n by n matrices with all entries equal to 1, find ( AB ) ij . 2. The same notation from part (a) turns the associative law ( AB ) C = A ( BC ) into X j X k a ik b kj ! c jl = X k a ik X j b kj c jl . Compute the value of both sides if C is also n by n , with every c jl = 2 ( A and B take the same values as in part 1). 3. For the matrix A = [ 1 2 1 0 0 4 ], extend the set of rows to a basis for < 3 , and (separately) reduce the set of columns to a basis for < 2 . 4. Prove that if V and W are 3-dimensional subspaces of < 5 , then V and W must have a nonzero vector in common. HINT: Start with bases for the two subspaces, making six vectors in all. Consider notions

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture