Math 21b: Linear Algebra Spring 2016

Homework 13: Gram Schmidt and QR

This homework is due on Friday, March 4, respectively on Tuesday, March 8, 2016.
   
5   1  

4   1 
   

1 Perform the Gram-Schmidt process on the two vectors {  ,  }.


3   2 




1 1
   

2 Find an orthonormal basis of the hyper plane x1 +x2 +x3 +x4 = 0

in R4.
3 As in the previous problem, we first find an orthonormal basis of
the kernel of  
 1 −1 1 −1 1 
A =  .
1 1 −1 1 1
Then find an orthonormal basis for the image.
4 Find the QR factorization of the following three matrices
   
0 

−3 0   1 
   
0 0 0   2   3 4 
   
   
A= ,B = 
 ,C = 
 .

 
7 0 0   3  −4 3
  

  
  
0 0 4 4
   

5 Find the QR factorization of the following matrices:

 
1 1 1   
 1 2 3 


1 1 −1 −1  

 
A=  0 5 6  ,
 
 
2 1 −1 1  

   
 0 0 6
 

1 1 −1

  
1 

0 0 0   
0 1 0 0   1 −1 
 
 
B=  ,C = 
 .

 
0 0 1 0  1 1
 
 


0 0 0 1
 

Gram Schmidt and QR

The Gram Schmidt orthogonalization process produces

from an arbitrary basis B = {vj } an orthonormal basis {uj }.
This goes as follows: w ~ 1 = ~v1 and ~u1 = w ~ 1|. To construct
~ 1/|w
ui once you’ve already constructed ~u1, . . . , ~ui−1 so that they
are orthonormal, make the new vector w ~ i = ~vi − projVi−1 (~vi),
where Vi−1 = span(u1, . . . , ui−1), and then normalize w ~ i to to
get ~ui = w ~ i|. Then {~u1, . . . , ~un} is an orthonormal basis of
~ i/|w
V and the formulas
~v1 = |~v1|~u1 = r11~u1
...

~vi = (~u1 · ~vi)~u1 + . + (~ui−1 · ~vi)~ui−1 + |w

~ i|~ui = r1i~u1 + .. + rii~ui
...

~vn = (~u1 ·~vn)~u1 +.+(~un−1 ·~vn)~un−1 +|w

~ n|~un = r1n~u1 +..+rnn~un
can be written in matrix form as A = QR,
    


| | |   | | | 



r11 r12 · · · r1n 

~v1 · · · ~vn  =  ~u1 · · · ~un 0 r22 · · · r2n ,
     
   
   
   
| | | | | | 0 0 · · · rnn
     

where A and Q are m × n matrices and R is an n × n matrix.