Mathematics 2270
2004
Homework 2-Sketch of solutions
2.1.2 The matrix is
Fall


0 2 0
0 0 3 
1 0 0
2.1.3 The map is not linear because we have
 
   
   
1
−1
−2
2
−2













2T
0
=2 1
=
2
and T
0
=
4
1
1
2
2
2
 
  
1
1
that is 2T 0 6= T 2 0.
1
1
2.1.5 The matrix is
7 6 −13
11 9 17
2.2.6 Let’s denote the projection by projL . This is a linear map from R3 to R3 . Following
the same calculations that for a projection in the plane we have:
projL (~x) = (~x.~u)~u
for ~u a unit vector parallel to L. We need then to find a unit vector parallel to L, it is
 
 
2 2
1 
1 because 1 = 3
~u =
3
2 2
 
x1
Then we have for ~x = x2  ∈ R3
x3
     
2
2
x1
1
1
projL (~x) = (~x.~u)~u = x2  . 1 1
3
3
2
2
x3
 
2
1

= (2x1 + x2 + 2x3 ) 1
9
2


4x + 2x2 + 4x3
1 1
2x1 + x2 + 2x3 
=
9
4x1 + 2x2 + 4x3
Then the matrix representing projL is


4 2 4
1
A = 2 1 2
9
4 2 4
2.2.7 Let ~x, ~x⊥ , ~x|| ∈ R3 such that ~x = ~x|| + ~x⊥ with ~x⊥ ⊥ L and ~x|| is parallel to L. We
know that then we have
refL (~x) = ~x|| − ~x⊥ = ~x|| − (~x − ~x|| ) = 2~x|| − ~x = 2 projL (~x) − ~x
Indeed we know that projL (~x) = ~x|| . Then
 

1
4
1
projL 1 = 2
9
1
4
Then we have
we use the previous exercises to compute
 
 
2 4
1
10
1
1 2  1  =  5 
9
2 4
1
10

   
1/9
1
10
1
refL (~x) =  5  − 1 = −4/9
9
1/9
1
10

2.2.15 We know (see previous exercises) that we have
refL (~x) = 2 projL (~x) − ~x
Moreover we know that (because ~u is a unit vector parallel to L):
  

u1
x1 u21 + x2 u1 u2 + x3 u1 u3
projL (~x) = (~x.~u)~u = (x1 u1 + x2 u2 + x3 u3 ) u2  = x1 u1 u2 + x2 u22 + x3 u2 u3 
u3
x1 u1 u3 + x2 u2 u3 + x3 u23
Then we have

   

x1 (u21 − 1) + x2 u1 u2 + x3 u1 u3
x1
x1 u21 + x2 u1 u2 + x3 u1 u3
refL (~x) = x1 u1 u2 + x2 u22 + x3 u2 u3  − x2  = x1 u1 u2 + x2 (u22 − 1) + x3 u2 u3 
x1 u1 u3 + x2 u2 u3 + x3 (u23 − 1)
x2
x1 u1 u3 + x2 u2 u3 + x3 u23
then the matrix of refL is:

 2
u1 u3
u1 − 1 u 1 u2
A =  u1 u2 u22 − 1 u2 u3 
u1 u3
u2 u3 u23 − 1