Download MATH 3328 Differential Equations

MATH 3323 Linear Algebra
Problem Set 6
Due April 21, 2011
On separate sheets of paper please solve all the problems below.
1. Find the domain and codomain of the following transformations and state whether
or not they are linear. If so, find the standard matrix for the transformation.
w1  2 x1  x 2
 2  1 0
3
3
a) w2  8 x1  3x3 R  R , linear, T   8 0 3


 1
0 0
w3  x1
b)
w1  x1  x 5 2
w2  x1 x3  x 4
R 4  R 2 , not linear
2.
a) Find the standard matrix for the linear transformation defined by the
0 1 2
formula T ( x1 , x2 , x3 )  ( x2  2 x3 ,2 x1  x2  x3 ) . T  

2  1 1 
b) Find T ( 2,1,3) by using the definition of T .
T (2,1,3)  (1  2(3), 2(2)  1  3)  (7,6)
 2
0 1 2   7
c) Find T ( 2,1,3) by using the standard matrix of T . Tx  
 1   
 2  1 1   3  6 
 
3. Given u  (1,4,3) . Use matrix multiplication to find the orthogonal projection of
u on the xz-plane, rotation of u about the z-axis by angle π/3, reflection of u about
x-axis followed by contraction by factor 1/5.
0  1 0 0  cos( / 3)  sin(  / 3) 0 1 0 0  1   1 / 10 
1 / 5 0

Tx   0 1 / 5 0  0  1 0   sin(  / 3) cos( / 3) 0 0 0 0  4    3 / 10
 0
0 1 / 5 0 0  1 
0
0
1 0 0 1  3    3 / 5 
4.
a) Find the standard matrix of the linear transformation “projection onto the
line y  2 x ”.
Standard matrix for projecting onto a line through the origin
 cos 2 x
cos x sin x
=

cos 2 x 
cos x sin x
If you draw this line, you can form a right triangle and determine that
cos x  1 / 5 and sin x  2 / 5 . Plugging this in the standard matrix
 1 / 5 2 / 5
becomes: 

 2 / 5 4 / 5
b) Use your answer in part a) to find the projection of the vector (5,-3) onto the
1 / 5 2 / 5  5    1 / 5 
line y  2 x . 
   

2 / 5 4 / 5  3  2 / 5
c) Check your work by calculation proj (1, 2) (5,3) .
(1,2)  (5,3)
1
 1  2 
(1,2) 
(1,2)   ,

5
5
 5 5 
5. Use theorem 4.3.3 to find the standard matrix for reflection about the xy-plane
followed by dilation by factor 3 followed by orthogonal projection onto the xzplane in R 3 (use standard basis vectors).
T  [T (e1 ) | T (e2 ) | T (e3 )]
proj (1, 2) (5,3) 
We take e1 , reflex about xy-plane, dilation by factor of 3, then orthogonal
projection onto xz-plane so
1
1
3
3






e1  0, T (e1 )  0  0  0
0
0
0
0
Do the same for e2 & e3 .
0
0 0
0






e2  1, T (e2 )  1  3  0
0
0 0
0
0
0
0
0






e3  0, T (e3 )   0    0    0 
1
 1
 3
 3
3 0 0 
So T  0 0 0 
0 0  3
6. Consider the bases B  [u1 , u2 ] and B'  [v1 , v2 ] for R 2 where
 2
4
  2
6
u1   , u 2   , v1   , v2   
 2
 1
3
1
a) Find the transition matrix from B’ to B.
v1  u1  u 2
1
1
 1 1
, [v1 ] B   , [v2 ] B    , P  

v 2  u1  u 2
 1
1
 1 1
b) Find the transition matrix from B to B’.
1 / 2  1 / 2
P 1  

1 / 2 1 / 2 
3
c) Compute the coordinate vector [w] B where [ w] B '   
 5
 1 1  3   2
 1 1  5    8

   
7. Determine if the following are one-to-one linear operators. If so, find the inverse
operator.
1 0 0 
3
d) Reflection about x-axis in R : T  0  1 0  det(T)=1 so T is invertible


0 0  1
and hence one to one. T
1
1 0 0 
 0  1 0 
0 0  1
4 0
e) Dilation by factor 4 in R 2 : T  
 det(T)=16 so T is invertible and hence
0 4
1 / 4 0 
one to one. T 1  

 0 1 / 4
0 0 
f) Orthogonal projection onto y-axis in R 2 : T  
 det(T)=0 so T is not
0 1 
invertible and hence not one to one.