mat 242 linear

MAT 242 Written Homework #5

EP5.1, 5.4 / H6.1–6.2 Due: November 17

Solve the following problems, showing any necessary work.

1. [1 point] Find a basis for W ⊥, the orthogonal complement of W , if W is the subspace spanned by

   

2 −2 −2 20

  ,

 

1 −3 0 5

  ,

  −3 1 −2 −5

   

2. Let ~v1 =

  −1 −1 1 1

 , ~v2 =

  −1 −1 −1 −1

 , and ~v3 =

  −1 1 1 −1

 . Note that B = {~v1,~v2,~v3} is an orthogonal set. Also,

let W be the subspace spanned by {~v1,~v2,~v3}.

a. [1 point] Find the vector in W closest to

  −1 −3 −7 −1

 , without inverting any matrices or solving any

systems of linear equations.

b. [1 point] Find an orthonormal basis for W .

3. Let W be the subspace spanned by

   

1 −2 0 2

  ,

  −1 2 1 −2

  ,

  −5 7 2 −4

   . Note that this basis is not orthogonal.

a. [1 point] Find the orthogonal projection of

 

7 −11 −3 −1

  into W .

b. [1 point] Find an orthogonal basis for W .