Linear Algebra

186 Chapter 3. Maps Between Spaces
This description of the projection in terms of shadows is is memorable, but
strictly speaking, R 2 isn’t equal to the xy-plane inside of R 3 (it is composed of
two-tall vectors, not three-tall vectors). Separating the two spaces by sliding
R 2 over to the right gives an instance of the general diagram above.
�w2
�w1
�w1 + �w2
The vectors that map to �w1 on the right have endpoints that lie in a vertical
line on the left. One such vector is shown, in gray. Call any such member
of the inverse image of �w1 a“�w1 vector”. Similarly, there is a vertical line of
“ �w2 vectors”, and a vertical line of “ �w1 + �w2 vectors”.
We are interested in π because it is a homomorphism. In terms of the
picture, this means that the classes add; any �w1 vector plus any �w2 vector
equals a �w1 + �w2 vector, simply because if π(�v1) = �w1 and π(�v2) = �w2 then
π(�v1 + �v2) =π(�v1) +π(�v2) = �w1 + �w2. (A similar statement holds about the
classes under scalar multiplication.) Thus, although the two spaces R 3 and R 2
are not isomorphic, π describes a way in which they are alike: vectors in R 3 add
like the associated vectors in R 2 —vectors add as their shadows add.
2.6 Example A homomorphism can be used to express an analogy between
spaces that is more subtle than the prior one. For instance, this map from R 2
to R 1 is a homomorphism.
� �
x h
↦−→ x + y
y
Fix two numbers a and b in the range R. Then the preservation of addition
condition says this for two vectors �u and �v from the domain.
� �
� �
u1
v1
if h( )=a and h( )=b then h(
u2
v2
� u1 + v1
u2 + v2
�
)=a + b
As in the prior example, we illustrate by showing the class of vectors in the
domain that map to a, the class of vectors that map to b, and the class of
vectors that map to a + b. Vectors that map to a have components that add
to a, so a vector is in the inverse image h −1 (a) if its endpoint lies on the line
x+y = a. We can call these the “a vectors”. Similarly, we have the “b vectors”,
etc. Now the addition preservation statement becomes this.