Linear Algebra with Applications (3rd Edition)

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
110.201 Linear Algebra 4th Quiz April 8, 2005 Problem 1 Find an orthogonal basis for the plane x - y + z = 0 , viewed as a subspace of R 3 . Solution So it is easy to see that the basis for the plane is v 1 = 1 1 0 , v 2 = 1 0 - 1 Now let’s use Gram-Schmidt algorithm to get the orthogonal basis of it. v 1 = 1 1 0 , then | v 1 | = v 1 · v 1 = 2 So we have u 1 = v 1 | v 1 | = 1 2 1 1 0 And v 2 = v 2 - Proj u 1 v 2 = v 2 - ( v 2 · u 1 ) u 1 = v 2 - 1 2 v 1 = 1 2 - 1 2 - 1 , and | v 2 | = 3 2 Therefore, u 2 = v 2 | v 2 | = 2 3 1 2 - 1 2 - 1 So the orthogonal basis for the plan is 1 2 1 1 0 , 2 3 1 2 - 1 2 - 1 1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Problem 2 Let e 1 , e 2 , e 3 be the standard basis of R 3 . Consider the plane V spanned by e 1 and e 2 . a. For a given vector w = ( a, b, c ) R 3 , calculate the vector u V that minimizes the distance between V and w , i.e. find u V such that u - w v - w v V. Solution u = Proj V w = a b 0 b. In the inequality above, is such a
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern