Transcript Chapter Three: Maps Between Spaces

Chapter Three: Maps Between Spaces
I.
II.
III.
IV.
V.
VI.
Isomorphisms
Homomorphisms
Computing Linear Maps
Matrix Operations
Change of Basis
Projection
Topics:
• Line of Best Fit
• Geometry of Linear Maps
• Markov Chains
• Orthonormal Matrices
3.I. Isomorphisms
3.I.1. Definition and Examples
3.I.2. Dimension Characterizes Isomorphism
3.I.1. Definition and Examples
Definition 1.3:
Isomorphism
An isomorphism between two vector spaces V and W is a map f : V →W that
(1)
is a correspondence: f is a bijection (1-to-1 and onto);
(2)
preserves structure:
f a   b   f a   f  b
 a, b V ,  ,   R
In which case, V  W, read “ V is isomorphic to W . ”
Example 1.1:
n-wide Row Vectors  n-tall Column Vectors
Example 1.2:
Pn  Rn+1
Example 1.4:
 c cos  c
Example 1.5:
c xc yc z
1
1
2
2
sin  c1 , c2  R 
3
  c1  
      R2
  c2  
c1 , c2 , c3  R    c1  c2 x  c3 x 2   P2
Automorphism
Automorphism = Isomorphism of a space with itself.
Example 1.6: Dilation :
d s : Rn  Rn by v
sv
Rotation :
t : Rn  Rn by v
R    v
Reflection :
fl : Rn  Rn by v
σl  v
Example 1.7: Translation
f : Pn  Pn by p  x 
Symmetry: Invariance under mapping.
p  x  a
Lemma 1.8:
Proof:
An isomorphism maps a zero vector to a zero vector.
0vV  0V
f 0vV   0 f  vV   0W
Lemma 1.9:
For any map f : V → W between vector spaces these statements are equivalent.
(1) f preserves structure
f(v1 + v2) = f(v1) + f(v2) and f(cv) = c f(v)
(2) f preserves linear combinations of two vectors
f(c1v1 + c2v2) = c1 f(v1) + c2 f(v2)
(3) f preserves linear combinations of any finite number of vectors
f(c1v1 +…+ cnvn) = c1 f(v1) +…+ cn f(vn)
Proof: See Hefferon p.175.
Exercises 3.I.1
1.
Show that the map f : R → R given by f(x) = x3 is one-to-one and onto.
Is it an isomorphism?
2. (a) Show that a function f : R2 → R2 is an automorphism iff it has the form
 ax  by 
 cx  d y 


 x
 y
 
where a, b, c, d  R and ad  bc  0
(b) Let f be an automorphism of R2 with
 1  
2
f      
 1
 3  
Find
 0 
f  
  1  
and
 1 
 0
f     
1
 4 
3. Show that, although R2 is not itself a subspace of R3, it is isomorphic to
the xy-plane subspace of R3.
4. Let U and W be vector spaces. Define a new vector space, consisting of
the set U  W = { ( u, w ) | u  U and w  W } along with these operations.
u1 , w1   u2 , w2   u1  u2 , w1  w2 
and
a  u , w   au , a w
This is a vector space, the external direct sum (Cartesian product) of U and W.
(a) Check that it is a vector space.
(b) Find a basis for, and the dimension of, the external direct sum P2  R2.
(c) What is the relationship among dim(U), dim(W), and dim(U  W)?
(d) Suppose that U and W are subspaces of a vector space V such that V = U  W
(in this case we say that V is the internal direct sum of U and W).
Show that the map f : U  W → V given by
( u, w )  u + w
is an isomorphism. Thus if the internal direct sum is defined then the internal
and external direct sums are isomorphic.
3.I.2. Dimension Characterizes Isomorphism
Theorem 2.1:
Isomorphism is an equivalence relation between vector spaces.
Proof: ( For details, see Hefferon p.179 )
1) Reflexivity: Identity map, id: v  v, preserves L.C.
2) Symmetry: f is bijection → f 1 exists & preserves L.C.
3) Transitivity: Composition preserves L.C.
Isomorphism classes:
Theorem 2.3:
Vector spaces are isomorphic  they have the same dim.
Proof: (see Hefferon p.180)
Isomorphism → correspondence between bases.
Lemma 2.4:
If spaces have the same dimension then they are isomorphic.
Proof: (see Hefferon p.180)
Every n-D vector space is isomorphic to Rn.
Decomposition
v  v1 β1 
 vn β n
 v1 
~  
v 
 n B
is unique for given B.
Isomorphism classes are characterized by dimension.
Corollary 2.6:
A finite-dimensional vector space is isomorphic to one and only one of the Rn.
Example 2.7:
M22  R4
B   β1 , β2 , β3 , β4  
 1 0  0 1  0 0  0 0
 0 0 ,  0 0 , 1 0 ,  0 1

 
 
 

E4   e1 , e 2 , e 3 , e 4  
1  0  0  0
 0 1  0  0
 , , , 
 0  0 1  0
       
 0  0  0 1
a b 
M
 a β1  b β2  c β3  d β4

c d 
a
a/2
b
b/2

M  
c
 c/2
 


d
d
/
2
 B 
D
where
a
b
a e1  b e 2  c e3  d e 4   
c
 
d 
D   2β1, 2β2 ,2β3,2β4

Exercises 3.I.2.
1. Consider the isomorphism RepB(·) : P1 → R2 where B =  1, 1+x .
Find the image of each of these elements of the domain.
(a) 3  2x;
(b) 2 + 2x;
(c) x
2. Suppose that V = V1  V2 and that V is isomorphic to the space U under the
map f. Show that U = f(V1)  f(V2).