Linear Algebra, 2020a

206 Chapter Three. Maps Between Spaces
2.15 Example Where h: R 3 → R 4 is
⎛ ⎞
⎛ ⎞ x
x
⎜ ⎟
⎝y⎠
↦−→
h
0
⎜ ⎟
⎝y⎠
z
0
the range space and null space are
⎛ ⎞
a
⎛
0
R(h) ={ ⎜ ⎟
⎝b⎠ | a, b ∈ R} and N (h) ={ ⎜
0
⎞
⎟
⎝0⎠ | z ∈ R}
z
0
and so the rank of h is 2 while the nullity is 1.
2.16 Example If t: R → R is the linear transformation x ↦→ −4x, then the range
is R(t) =R. The rank is 1 and the nullity is 0.
2.17 Corollary The rank of a linear map is less than or equal to the dimension of
the domain. Equality holds if and only if the nullity of the map is 0.
We know that an isomorphism exists between two spaces if and only if the
dimension of the range equals the dimension of the domain. We have now seen
that for a homomorphism to exist a necessary condition is that the dimension of
the range must be less than or equal to the dimension of the domain. For instance,
there is no homomorphism from R 2 onto R 3 . There are many homomorphisms
from R 2 into R 3 , but none onto.
The range space of a linear map can be of dimension strictly less than the
dimension of the domain and so linearly independent sets in the domain may
map to linearly dependent sets in the range. (Example 2.3’s derivative transformation
on P 3 has a domain of dimension 4 but a range of dimension 3 and the
derivative sends {1, x, x 2 ,x 3 } to {0, 1, 2x, 3x 2 }). That is, under a homomorphism
independence may be lost. In contrast, dependence stays.
2.18 Lemma Under a linear map, the image of a linearly dependent set is linearly
dependent.
Proof Suppose that c 1 ⃗v 1 + ···+ c n ⃗v n = ⃗0 V with some c i nonzero. Apply h to
both sides: h(c 1 ⃗v 1 + ···+ c n ⃗v n )=c 1 h(⃗v 1 )+···+ c n h(⃗v n ) and h(⃗0 V )=⃗0 W .
Thus we have c 1 h(⃗v 1 )+···+ c n h(⃗v n )=⃗0 W with some c i nonzero. QED
When is independence not lost? The obvious sufficient condition is when
the homomorphism is an isomorphism. This condition is also necessary; see