Chapter Three: Maps Between Spaces
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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 xc yc 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:
M22 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).