Download Constructions in linear algebra For all that follows, let k be the base

Constructions in linear algebra
For all that follows, let k be the base field.
1. Let e1 , . . . , en be a basis for V . Define
fi (a1 e1 + · · · + an en ) = ai
Show that f1 , . . . , fn is a basis for V ∗ . This is the canonical basis dual to e1 , . . . , en .
2. Let V be a finite-dimensional vector space, and let V ∗ be its dual. For v ∈ V and f ∈ V ∗ ,
let the symbol f ⊗ v denote the endomorphism, i.e. linear transformation V → V , defined by
(f ⊗ v)(x) = f (x) · v
where · denotes scalar multiplication. Show that
{f ⊗ v | v ∈ V, f ∈ V ∗ }
spans End(V ), the vector space of linear transformations V → V .
3. What is the rank of f ⊗ v?
4. As above, suppose that e1 , . . . , en is a basis for V and f1 , . . . , fn is a basis for V ∗ . Find a
basis of End(V ). What is the dimension of End(V )?
5. Suppose that V is infinite-dimensional. Do the symbols f ⊗ v span End(V )?
6. Let evv : V ∗ → k denote the “evaluation at v” map. Define the map f : V → V ∗∗ by v 7→ evv .
Show that f is injective (do not assume that V is finite-dimensional).
7. Give an explicit example for when f is not an isomorphism.
8. Let W ⊂ V . Define the orthogonal space to W by
W ⊥ := {f ∈ V ∗ | f (w) = 0 ∀w ∈ W }
Show that W ⊥ is a subspace of V ∗ . Compute its dimension in terms of dim(W ) and dim(V ).
9. Let f : V → W be a linear transformation. Show that there is a canonical linear transformation f ∗ : W ∗ → V ∗ . Hint: it is defined by f ∗ (g) = g ◦ f .
The following problems are harder, but they cover a lot of important constructions in linear algebra.
You will need to know what an equivalence relation is.
1. Let W ⊂ V . Define the quotient space V /W to be the set of equivalence classes under
the equivalence relation v ∼ v 0 if and only if v − v 0 ∈ W . Define an addition and scalar
multiplication on V /W and show that it is a vector space. Furthermore, show that the
quotient map V → V /W is a linear transformation.
2. Let W ⊂ V . Note that there is no canonical inclusion W ∗ ⊂ V ∗ . Let i : W → V be the
inclusion; then there is a map i∗ : V ∗ → W ∗ . Show that i∗ is a quotient map, and show that
its kernel is W ⊥ .
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Constructions in linear algebra
3. Let S be a set. The free vector space on the set S, denoted kS, is defined to be the subset of
all functions (not linear! S is an arbitrary set) f : S → k such that f is zero on a cofinite set
(i.e. f is zero except on a finite set). Equivalently, kS is the set of sequences indexed by S
where cofinitely many terms are zero; addition and scalar multiplication are component wise.
Again equivalently, kS is the set of formal sums of elements of S with coefficients in k. We
often write,
M
kS =
k
s∈S
Pick your favorite characterization of kS and prove it is a vector space. If this is confusing
to you, consider the next problem first.
4. Let S = {♠, ♣}. Elements of kS have the form a♠ + b♣ where a, b ∈ k. Show that kS
is two-dimensional. Show in general that if S is a finite set with n elements, then kS is
n-dimensional.
5. Define the tensor product V ⊗W of two vector spaces V, W as the quotient k(V ×W )/ ∼ where
v ⊗w +v 0 ⊗w ' (v +v 0 )⊗w, v ⊗w +v ⊗w0 ∼ v ⊗(w +w0 ), and k(v ⊗w) ∼ (kv)⊗w ∼ v ⊗(kw).
Let v1 , . . . , vn be a basis for V and w1 , . . . , wm a basis for W . Show that {vi ⊗vj | i ∈ [1, n], j ∈
[1, m]} is a basis for V ⊗ W . Conclude that dim(V ⊗ W ) = dim(V ) dim(W ).
6. Show that for V finite-dimensional, there is a natural isomorphism End(V ) ' V ∗ ⊗ V . When
V is not finite-dimensional, show that there is a natural map V ∗ ⊗ V → End(V ), which is
injective, but might not be surjective. Hint: you already did this; it’s the first few problems
above.
7. Suppose that V, W are finite-dimensional. Let f : V → W . Show that this gives us natural
associated maps: V ⊗ W ∗ → k and k → V ⊗ W ∗ .
8. Suppose V is finite-dimensional. Let k → V ∗ ⊗ V be the coevaluation map η defined by
1 7→
n
X
fi ⊗ ei
i=1
where e1 , . . . , en is a choice of basis for V , and f1 , . . . , fn is the corresponding dual basis
(which is uniquely determined). Show that the coevaluation map is independent of the choice
of basis ei . Show that, if we identify V ∗ ⊗ V ' End(V ), that the coevaluation sends a ∈ k
to aI, i.e. a times the identity transformation. (When V is infinite-dimensional, this map
k → End(V ) is still well-defined, while k → V ∗ ⊗ V is not.)
9. Define the evaluation map : V ∗ ⊗ V → k by
(f1 ⊗ v1 + · · · + fr ⊗ vr ) = f1 (v1 ) + · · · fr (vr )
Show that this map is well-defined (for example, show that f (v ⊗w +v ⊗w0 ) = f (v ⊗(w +w0 )),
and do the same for the other relations). Note that we don’t have to assume that V is finitedimensional.
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Constructions in linear algebra
10. Show that if V is finite-dimensional, then ◦η : k → k is the map which multiplies by dim(V ).
Note that if V is infinite-dimensional, we can’t define this composition, since when we try to
do a composition, the middle map
k → End(V ) ← V ∗ ⊗ V → k
goes the “wrong way.”
11. Let h−, −i be an inner product on finite-dimensional V . Express the inner product as a map
V ⊗ V → k. Use the above exercises to show that an inner product defines an isomorphism
V ' V ∗ . Hint: any bilinear form defines a map V ⊗ V → k, and thus a map V → V ∗ . The
inner product is “nondegenerate,” meaning this map is an isomorphism.
12. An inner product on a finite-dimensional vector space also gives us a map k → V ∗ ⊗ V ∗ .
In particular, evaluating this map at 1 ∈ k, we obtain an element of V ∗ ⊗ V ∗ , so the inner
product is an element of V ∗ ⊗ V ∗ . The vector space V ∗ ⊗ V ∗ is the space of bilinear forms;
is the subset of inner products a vector subspace of V ∗ ⊗ V ∗ ?
Finally, here is an aside on direct sums. It may be confusing because of overlap between the book’s
terminology and what I consider more standard.
1. Ignore what your book says about direct sums. Let V, W be two vector spaces. The direct
sum V ⊕ W is the vector space whose underlying set is V × W , where addition and scalar
multiplication are defined component-wise. Show that this is a vector space.
2. Let ιV : V → V ⊕ W be defined by ιV (v) = (v, 0), and likewise for ιW . Show that ιV , ιW are
injective.
3. Show that the images (ranges) of ιV , ιW are subspaces of V ⊕ W with empty intersection, and
such that range(ιV ) + range(ιW ) = V ⊕ W . Relate this to the book’s notion of direct sum.
4. Show that if dim(V ) = n and dim(W ) = m, then dim(V ⊕ W ) = n + m.
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