Algebra Written HW

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