# 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.

**. If then is a subspace if and only if or .**

Exercise 2.7

Exercise 2.7

**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

**We denote the**

Definition 2.30.

Definition 2.30.

**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.

**If U ≤ V and A is a basis of U then A can be extended to a basis of V .**

Lemma 2.33.

Lemma 2.33.

**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.

**Show that if F,G are number fields and F G then G is a vector space over**

Exercise 2.40.

Exercise 2.40.

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 square-free 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 row-operation**is an elementary operation applied

to the rows of the matrix. Elementary column operations are defined analogously. Exercise ??

shows that

**elementary row-operations**do not change the

**row-rank**of A.

**Exercise 2.53.**Show that elementary

**row-operations**do not change the

**column-rank**of

a matrix.

**Exercise 2.54.**Use Exercises ?? and ?? prove the "amazing" Theorem ??.

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