Solve the following problems, showing any necessary work. Your answers must be exact, or must contain atleast four digits after the decimal point. MAT 242 Written Homework #5

6.1–6.3

Due: November 22

Solve the following problems, showing any necessary work. Your answers must be exact, or must contain at

least four digits after the decimal point.

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

2

−2

5

5 2 −3

,

,

3

0

2

−6

4

10

2

−6

−15

0

−7

14 →

→

−

→

−

→

−

→

−

−

→

−

2. Let v1 =

, let B = (v1 , v2 , v3 ), and let W be the subspace

, and v3 =

, v2 =

−3

16

−2

−6

−10

−4

spanned by B. Note that B is an orthogonal set.

−49

35

→

−

a. [1 point] Find the coordinates of u =

with respect to B, without inverting any matrices or

7

−28

solving any systems of linear equations.

−28

−35

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

, without inverting any matrices or solving any

35

−35

systems of linear equations.

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

is an orthonormal basis for W.

3. [1 point] Find the Least Squares Solution to the following system of linear equations:

x − y + 2z =

2

−2 x + 3y − 6z = −10

2x

− z = −5

2 x − y + 3z = −7

2z = −10

x − y + 4z = −8