Linear Algebra, 2020a

428 Chapter Five. Similarity
(c) t 2 : P 2 → P 2 , a + bx + cx 2 ↦→ b + cx + ax 2
(d) t 3 : R 3 → R 3 ,
⎛ ⎞ ⎛ ⎞
a a
⎝b⎠ ↦→ ⎝a⎠
c b
1.11 Prove that function composition is associative (t ◦ t) ◦ t = t ◦ (t ◦ t) and so we
can write t 3 without specifying a grouping.
1.12 Check that a subspace must be of dimension less than or equal to the dimension
of its superspace. Check that if the subspace is proper (the subspace does not equal
the superspace) then the dimension is strictly less. (This is used in the proof of
Lemma 1.4.)
̌ 1.13 Prove that the generalized range space R ∞ (t) is the entire space, and the
generalized null space N ∞ (t) is trivial, if the transformation t is nonsingular. Is
this ‘only if’ also?
1.14 Verify the null space half of Lemma 1.4.
̌ 1.15 Give an example of a transformation on a three dimensional space whose range
has dimension two. What is its null space? Iterate your example until the range
space and null space stabilize.
1.16 Show that the range space and null space of a linear transformation need not
be disjoint. Are they ever disjoint?
III.2
Strings
This requires material from the optional Combining Subspaces subsection.
The prior subsection shows that as j increases the dimensions of the R(t j )’s
fall while the dimensions of the N (t j )’s rise, in such a way that this rank and
nullity split between them the dimension of V. Can we say more; do the two
split a basis — is V = R(t j ) ⊕ N (t j )?
The answer is yes for the smallest power j = 0 since V = R(t 0 ) ⊕ N (t 0 )=
V ⊕ {⃗0}. The answer is also yes at the other extreme.
2.1 Lemma For any linear t: V → V the function t: R ∞ (t) → R ∞ (t) is one-toone.
Proof Let the dimension of V be n. Because R(t n )=R(t n+1 ), the map
t: R ∞ (t) → R ∞ (t) is a dimension-preserving homomorphism. Therefore, by
Theorem Three.II.2.20 it is one-to-one.
QED