Linear Algebra, 2020a

180 Chapter Three. Maps Between Spaces
to break the sum along the final ‘+’.
f(c 1 ⃗v 1 + ···+ c k ⃗v k + c k+1 ⃗v k+1 )=f(c 1 ⃗v 1 + ···+ c k ⃗v k )+f(c k+1 ⃗v k+1 )
Use the inductive hypothesis to break up the k-term sum on the left.
Now the second half of (1) gives
when applied k + 1 times.
= f(c 1 ⃗v 1 )+···+ f(c k ⃗v k )+f(c k+1 ⃗v k+1 )
= c 1 f(⃗v 1 )+···+ c k f(⃗v k )+c k+1 f(⃗v k+1 )
QED
We often use item (2) to simplify the verification that a map preserves structure.
Finally, a summary. In the prior chapter, after giving the definition of a
vector space, we looked at examples and noted that some spaces seemed to be
essentially the same as others. Here we have defined the relation ‘ ∼ =’ and have
argued that it is the right way to precisely say what we mean by “the same”
because it preserves the features of interest in a vector space — in particular, it
preserves linear combinations. In the next section we will show that isomorphism
is an equivalence relation and so partitions the collection of vector spaces.
Exercises
̌ 1.12 Verify, using Example 1.4 as a model, that the two correspondences given before
the definition are isomorphisms.
(a) Example 1.1 (b) Example 1.2
̌ 1.13 For the map f: P 1 → R 2 given by
( )
a + bx ↦−→
f a − b
b
Find the image of each of these elements of the domain.
(a) 3 − 2x (b) 2 + 2x (c) x
Show that this map is an isomorphism.
1.14 Show that the natural map f 1 from Example 1.5 is an isomorphism.
1.15 Show that the map t: P 2 → P 2 given by t(ax 2 + bx + c) =bx 2 −(a + c)x + a is
an isomorphism.
̌ 1.16 Verify that this map is an isomorphism: h: R 4 → M 2×2 given by
⎛ ⎞
a
( )
⎜b
⎟ c a+ d
⎝c⎠ ↦→ b d
d
̌ 1.17 Decide whether each map is an isomorphism. If it is an isomorphism then prove
it and if it isn’t then state a condition that it fails to satisfy.
(a) f: M 2×2 → R given by
( ) a b
↦→ ad − bc
c d