Linear Algebra with Applications (3rd Edition)

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

110.201 Linear Algebra 4th Quiz April 8, 2005 Problem 1 Let e i , i = 1 , 2 , 3 , 4 be the standard basis of R 4 . Is there an orthogonal matrix A with Ae 1 = 1 2 0 0 1 2 , Ae 2 = 1 2 0 0 - 1 2 , Ae 3 = 0 1 0 0 , Ae 4 = 0 0 2 0 or not? If so, find A , and if not, explain. Solution By the definition of orthogonal matrix, we know | Ae 4 | = | e 4 | But | Ae 4 | = 2 while | e 4 | = 1 Therefore, the orthogonal matrix can not exist. Problem 2 Find an orthonormal basis for the subspace V R 3 spanned by the vectors 1 0 1 , 1 1 0 . Moreover, find a normal basis for the orthogonal complement of V . Solutions We use Gram-Schmidt to do this problem. v 1 = 1 0 1 1
so, u 1 = v 1 | v 1 | = 1 2 1 0 1 then, v 2 = v 2 - ( v 2 · u 1 ) u 1 = 1 1 0 - 1 2 1 0 1 = 1 2 1 - 1 2 Therefore, u 2 = v 2 | v 2 | = 2 3 1 2 1 - 1 2 The orthonormal basis of V is 1 2

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture