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Back to Finding orthogonal bases

Exercise 1

Avg difficulty 2.5

Suppose that a subspace WW of R3\real^3 has a basis formed by v1=[111],v2=[1−2−2].\vvec_1=\threevec111, \hspace{24pt} \vvec_2=\threevec1{-2}{-2}.

  1. Find an orthogonal basis for WW.

  2. Find an orthonormal basis for WW.

  3. Find the matrix PP that projects vectors orthogonally onto WW.

  4. Find the orthogonal projection of [34−2]\threevec34{-2} onto WW.

Austin, Understanding Linear Algebra, §6.4 Finding orthogonal bases, Exercise 1

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.