201 Quiz 4 TH solutions

Linear Algebra with Applications (3rd Edition)

Info iconThis 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 | = r 3 2 Therefore, u 2 = v 2 | v 2 | = r 2 3 1 2 - 1 2 - 1 So the orthogonal basis for the plan is 1 2 1 1 0 , r 2 3 1 2 - 1 2 - 1 1
Background image of page 1

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

View Full DocumentRight 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 k ~u - ~w k ≤ k ~v - ~w k ~v V. Solution ~u = Proj V w = a b 0 b. In the inequality above, is such a
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

201 Quiz 4 TH solutions - 110.201 Linear Algebra 4th Quiz...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online