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Linear Algebra
1 Subspaces
Definition 2.1. A subset U V is a
subspace (written U ≤ V ) if
Exercise 2.2. Show , Span(S) ≤
V , i.e. that Span(S) is a subspace of V . (Recall
that .)
Exercise 2.3. Show that Span(S) is the smallest subspace containing S,
i.e.
(a) Span(S) ≤ V and Span(S) S:
(b) If W ≤ V and S
W then Span(S) ≤
W:
Corollary 2.4. .
Exercise 2.5. Show that the intersection of any set of subspaces is a
subspace.
Remark 2.6. Note that this doesn't hold true for unions.
Exercise 2.7. If then
is a subspace if and only if
or .
2 All bases are equal
Theorem 2.8 (Fundamental Fact of Linear Algebra ). If L ,M
V with L is a
linearly
independent set of vectors, and L ≤ Span(M) then lLl ≤ lMl.
Lemma 2.9 (Steinitz Exchange Principle). If
are linearly independent and
then , 1≤
j ≤ l such that are linearly indepen 
dent. (Note in particular that .)
Exercise 2.10. Prove the Steinintz Exchange Principle.
Exercise 2.11. Prove the Fundamental Fact using the Steinitz Echange
Principle.
Definition 2.12. An m × n matrix is an m × n array of numbers
which we write as
Definition 2.13. The row (respectively column)
rank of a matrix is the rank of the set
of row (respectively column) vectors.
Theorem 2.14 (The Most Amazing Fact of Basic Linear Algebra ). The
row rank of
a matrix is equal to its column rank. (To be proven later in today.)
Definition 2.15. Let S
V , then a subset B
S is a basis
of S if
(a) B is linearly independent.
(b) S ≤ Span(B).
Exercise 2.16. Show that the statement: "All bases for S have
equal size" is equivalent
to Theorem ??.
Definition 2.17. We call the common size of all bases of S the rank
of S, denoted rk(S).
3 Coordinates
Exercise 2.18. Show that B
S is a basis if and
only if B is a maximal linearly independent
subset of S.
Exercise 2.19. If B is a basis of S then
there exists a unique linear combination of
elements in B that sums to x . In other words for all x there are unique scalars
such that
Definition 2.20. For a basis B, regarded as an
ordered set of vectors, we associate to each
x ∈S the column vector
called the coordinates of x , where the
are as above.
4 Linear maps, isomorphism of vector spaces
Definition 2.21. Let V and W be vector spaces. We say that a map f : V →
W is a
homomorphism or a linear map if
Exercise 2.22. Show that if f is a linear map then f(0) = 0.
Exercise 2.23. Show that .
Definition 2.24. We say that f is an isomorphism if f is a
bijective homomorphism.
Definition 2.25. Two spaces V and W are isomorphic if there exists
an isomorphism be
tween them.
Exercise 2.26. Show the relation of being isomorphic is an equivalence
relation.
Exercise 2.27. Show that an isomorphism maps bases to bases.
Theorem 2.28. If dim(V ) = n then .
Proof: Choose a basis, B of V , now map each vector to its coordinate
vector, i.e. v → [v]B.
Definition 2.29. We denote the image of f as the set
Definition 2.30. We denote the kernel of f as the set
Exercise 2.31. For a linear map f : V → W show that im(f) ≤ W and
ker(f) ≤ V .
Theorem 2.32. For a linear map f : V → W we have
dim ker(f) + dim im(f) = dim V.
Lemma 2.33. If U ≤ V and A is a basis of U then A can be extended to a
basis of V .
Exercise 2.34. Prove Theorem ??. Hint: apply Lemma ?? setting U = ker(f).
5 Vector spaces over number fields
Definition 2.35. A subset F
C is a number
field if F is closed under the four arithmetic
operations, i.e. for ,
Exercise 2.36. Show that if F is a number field then Q
F.
Exercise 2.37. Show that is a number
field.
Exercise 2.38. Show that is a number
field.
Exercise 2.39 (Vector Spaces over Number Fields). Convince yourself that
all of the
things we have said about vector spaces remain valid if we replace R and F.
Exercise 2.40. Show that if F,G are number fields and F
G then G is a vector space over
F.
Exercise 2.41. Show that .
Exercise 2.42. Show that has the
cardinality of "continuum," that is, it has the same
cardinality as R.
Exercise 2.43 (Cauchy' s Equation ). We consider functions f : R → R
satisfying Cauchy's
Equation : f(x + y) = f(x) + f(y) with x, y ∈ R. For such a function prove
that
(a) If f is continuous then f(x) = cx.
(b) If f is continuous at a point then f(x) = cx.
(c) If f is bounded on some interval then f(x) = cx.
(d) If f is measurable in some interval then f(x) = cx.
(e) There exists a g : R → R such that g(x) ≠ cx but g(x + y) = g(x) + g(y).
(Hint: Use
the fact that R is a vector space over Q. Use a basis of this vector space. Such
a basis
is called a Hamel basis.
Exercise 2.44. Show that 1,, and
are linearly independent over Q.
Exercise 2.45. Show that and
are linearly independent over
Q.
Exercise 2.46. *Show that the set of square roots
of all of the squarefree integers are linearly
independent over Q. (An integer is square free if it is not divisible by
the square of any prime
number. For instance, 30 is square free but 18 is not.)
Definition 2.47. A rational function f over R is a fraction of the
form
where g, h ∈ R[x] (that is g, h are real polynomials ) and h(x) ≠ 0 (h is
not the identically
zero polynomial ). More precisely , a rational function is an equivalence class of
fractions of
polynomials , where the fractions and
are equivalent if and only if
.
(This is analogous to the way fractions of integers represent rational numbers,
the fractions
3/2 and 6/4 represent the same rational number.) We denote the set of all
rational functions
as R(x).
Note that a rational function is not a function, it is an equivalence class of
formal quotients.
Exercise 2.48. Prove that the rational functions
are linearly independent set
over R(x).
Corollary 2.49. has the cardinality
of "conyinuum" (the same cardinality as R).
6 Elementary operations
Definition 2.50. The following actions on a set of vectors
are called elementary
operations:
(a) Replace by
where i ≠ j.
(b) Replace by
where α ≠ 0.
(c) Switch and
.
Exercise 2.51. Show that the rank of a list of vectors doesn't change
under elementary
operations.
Exercise 2.52. Let have rank r. Show
that by a sequence of elementary operations
we can get from to a set
such that
are linearly independent
and .
Consider a matrix. An elementary rowoperation is an elementary operation
applied
to the rows of the matrix. Elementary column operations are defined analogously.
Exercise ??
shows that elementary rowoperations do not change the rowrank of
A.
Exercise 2.53. Show that elementary rowoperations do not change
the columnrank of
a matrix.
Exercise 2.54. Use Exercises ?? and ?? prove the "amazing" Theorem ??.
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