Linear Algebra

Section II. Homomorphisms 193
2.38 Prove that the image of a span equals the span of the images. That is, where
h: V → W is linear, prove that if S is a subset of V then h([S]) equals [h(S)]. This
generalizes Lemma 2.1 since it shows that if U is any subspace of V then its image
{h(�u) � � �u ∈ U} is a subspace of W , because the span of the set U is U.
� 2.39 (a) Prove that for any linear map h: V → W and any �w ∈ W , the set
h −1 ( �w) has the form
{�v + �n � � �n ∈ N (h)}
for �v ∈ V with h(�v) = �w (if h is not onto then this set may be empty). Such a
set is a coset of N (h) and is denoted �v + N (h).
(b) Consider the map t: R 2 → R 2 � �
given
�
by
�
x t ax + by
↦−→
y cx + dy
for some scalars a, b, c, andd. Provethattis linear.
(c) Conclude from the prior two items that for any linear system of the form
ax + by = e
cx + dy = f
the solution set can be written (the vectors are members of R 2 )
{�p + �h � � �h satisfies the associated homogeneous system}
where �p is a particular solution of that linear system (if there is no particular
solution then the above set is empty).
(d) Show that this map h: R n → R m ⎛ ⎞ ⎛
is linear
⎞
⎜
⎝
x1
.
⎟
⎠ ↦→
⎜
⎝
a1,1x1 + ···+ a1,nxn
xn am,1x1 + ···+ am,nxn
for any scalars a1,1, ... , am,n. Extend the conclusion made in the prior item.
(e) Show that the k-th derivative map is a linear transformation of Pn for each
k. Prove that this map is a linear transformation of that space
f ↦→ dk d
f + ck−1
dxk k−1
d
f + ···+ c1 f + c0f
dxk−1 dx
for any scalars ck, ... , c0. Draw a conclusion as above.
2.40 Prove that for any transformation t: V → V that is rank one, the map given
by composing the operator with itself t ◦ t: V → V satisfies t ◦ t = r · t for some
real number r.
2.41 Show that for any space V of dimension n, thedual space
L(V,R) ={h: V → R � � h is linear}
is isomorphic to R n . It is often denoted V ∗ . Conclude that V ∗ ∼ = V .
2.42 Show that any linear map is the sum of maps of rank one.
2.43 Is ‘is homomorphic to’ an equivalence relation? (Hint: the difficulty is to
decide on an appropriate meaning for the quoted phrase.)
2.44 Show that the rangespaces and nullspaces of powers of linear maps t: V → V
form descending
V ⊇ R(t) ⊇ R(t 2 ) ⊇ ...
and ascending
{�0} ⊆N (t) ⊆ N (t 2 ) ⊆ ...
chains. Also show that if k is such that R(t k ) = R(t k+1 ) then all following
rangespaces are equal: R(t k )=R(t k+1 )=R(t k+2 ) .... Similarly, if N (t k )=
N (t k+1 )thenN (t k )=N (t k+1 )=N (t k+2 )=....
.
⎟
⎠