Fundamental Structures

1.1 Fields

Fields generalize the properties of familiar “continuous” number systems like the real numbers $ℝ$ and complex numbers $ℂ$ into arbitrary sets which follow a set of field axioms.

In an abstract sense, we need three “things” to define a field: a set of objects called scalars and two binary operations acting on those scalars.

Definition 1: Binary Operation

A function $𝑓$ is a binary operation on a set $𝑆$ iff:

𝑓 : 𝑆 \times 𝑆 \to 𝑆 .

This implicitly requires that any binary operation $𝑓$ on $𝑆$ is closed, meaning that applying $𝑓$ to any two elements of $𝑆$ results in another element of $𝑆$ .

Definition 2: Field

A field $(𝐹, +, \cdot)$ is a set $𝐹$ together with two binary operations $+$ (called addition) and $\cdot$ (called multiplication) such that the following properties hold:

Associativity of addition. $\forall 𝑎, 𝑏, 𝑐 \in 𝐹, (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐) .$
Associativity of multiplication. $\forall 𝑎, 𝑏, 𝑐 \in 𝐹, (𝑎 \cdot 𝑏) \cdot 𝑐 = 𝑎 \cdot (𝑏 \cdot 𝑐) .$
Commutativity of addition. $\forall 𝑎, 𝑏 \in 𝐹, 𝑎 + 𝑏 = 𝑏 + 𝑎 .$
Commutativity of multiplication. $\forall 𝑎, 𝑏 \in 𝐹, 𝑎 \cdot 𝑏 = 𝑏 \cdot 𝑎 .$
Existence of additive identity. $\exists 0 \in 𝐹 s.t. \forall 𝑎 \in 𝐹, 𝑎 + 0 = 𝑎 .$
Existence of multiplicative identity. $\exists 1 \in 𝐹 s.t. \forall 𝑎 \in 𝐹, 𝑎 \cdot 1 = 𝑎 .$
Existence of additive inverses. $\forall 𝑎 \in 𝐹, \exists (- 𝑎) \in 𝐹 s.t. 𝑎 + (- 𝑎) = 0 .$
Existence of multiplicative inverses. $\forall 𝑎 \in 𝐹, 𝑎 \neq 0 \Rightarrow \exists 𝑎^{- 1} \in 𝐹 s.t. 𝑎 \cdot 𝑎^{- 1} = 1 .$
Distributivity over addition. $\forall 𝑎, 𝑏, 𝑐 \in 𝐹, 𝑎 \cdot (𝑏 + 𝑐) = 𝑎 \cdot 𝑏 + 𝑎 \cdot 𝑐 .$

Together, these are called the field axioms. The elements of $𝐹$ are called $𝑭$ -scalars, and with relevant context, $𝐹$ -scalars may be referred to as simply scalars.

For any $𝑎 \in 𝐹$ , the element $- 𝑎$ is called the additive inverse or the negative of $𝑎$ . For any $𝑎 \in 𝐹 \ {0}$ , the element $𝑎^{- 1}$ is called the multiplicative inverse of $𝑎$ .

Notation

Often, the binary operations associated with a field are not explicitly listed. A field $(𝐹, +, \cdot)$ is usually denoted simply as $𝐹$ , where the operations of addition and multiplication are implied. The notation $+_{𝐹}$ and $\cdot_{𝐹}$ may be used to refer to the addition and multiplication operations of $𝐹$ when there is ambiguity.

Notation

The additive and multiplicative identities of a field $𝐹$ may be denoted $0_{𝐹}$ and $1_{𝐹}$ , respectively, when there is ambiguity.

Notation

When unambiguous, field multiplication can be denoted by juxtaposition. For example, $𝑎 \cdot 𝑏$ may be written as $𝑎 𝑏$ .

Property: Additive and Multiplicative Identities are Unique

Let $𝐹$ be a field. Then there is exactly one additive identity in $𝐹$ , and exactly one multiplicative identity in $𝐹$ . That is, $\exists! 0 \in 𝐹$ and $\exists! 1 \in 𝐹$ .

Proof: Additive and Multiplicative Identities are Unique.

Let $𝐹$ be a field, and suppose that $0_{1}$ and $0_{2}$ are both additive identities of $𝐹$ . By the definition of additive identity, we have $0_{1} + 0_{2} = 0_{1}$ and $0_{1} + 0_{2} = 0_{2}$ . Thus, $0_{1} = 0_{2}$ , so there is exactly one additive identity in $𝐹$ .

Suppose that $1_{1}$ and $1_{2}$ are both multiplicative identities of $𝐹$ . By the definition of multiplicative identity, we have $1_{1} \cdot 1_{2} = 1_{1}$ and $1_{1} \cdot 1_{2} = 1_{2}$ . Thus, $1_{1} = 1_{2}$ , so there is exactly one multiplicative identity in $𝐹$ . ∎

Property: Additive and Multiplicative Inverses are Unique

Let $𝐹$ be a field, and let $𝑎 \in 𝐹$ . Then there is exactly one additive inverse of $𝑎$ in $𝐹$ . If $𝑎 \neq 0$ , then there is exactly one multiplicative inverse of $𝑎$ in $𝐹$ .

That is, $\exists! (- 𝑎) \in 𝐹$ and $𝑎 \neq 0 \Rightarrow \exists! 𝑎^{- 1} \in 𝐹$ .

Definition 3: Field Subtraction

Let $𝐹$ be a field. The binary operation of subtraction $-$ on $𝐹$ is defined as follows:

\forall 𝑎, 𝑏 \in 𝐹, 𝑎 - 𝑏 = 𝑎 + (- 𝑏) .

The quantity $𝑎 - 𝑏$ is called the difference of $𝑎$ and $𝑏$ .

Definition 4: Field Division

Let $𝐹$ be a field. The binary operation of division $/$ on $𝐹$ is defined as follows:

\forall 𝑎, 𝑏 \in 𝐹, 𝑎 / 𝑏 = 𝑎 \cdot 𝑏^{- 1} .

The quantity $𝑎 / 𝑏$ is called the quotient of $𝑎$ and $𝑏$ .

Notation

Field division can be denoted by the fraction notation $\div$ . For example, $𝑎 / 𝑏$ may be written as $\frac{𝑎}{𝑏}$ .

Examples and Properties of Fields

Theorem 1

The set of all real numbers $ℝ$ is a field.

Proof: Theorem 1.

Let $𝑎, 𝑏, 𝑐$ be arbitrary elements of $ℝ$ . Choose $0 \in ℝ$ to be the additive identity and $1 \in ℝ$ to be the multiplicative identity.

Addition and multiplication of real numbers are associative and commutative, so the first four field axioms are satisfied.

The additive inverse of $𝑎$ is $- 𝑎$ , which is also a real number, so the axiom of existence of additive inverses is satisfied.

If $𝑎 \neq 0$ , choose the multiplicative inverse of $𝑎$ to be $1 / 𝑎$ , which is also a real number, so the axiom of existence of multiplicative inverses is satisfied.

We have $𝑎 (𝑏 + 𝑐) = 𝑎 𝑏 + 𝑎 𝑐$ , so the axiom of distributivity over addition is satisfied. Thus, all field axioms are satisfied, and $ℝ$ is a field. ∎

Proposition 1

The set of all complex numbers $ℂ$ is a field.

Proposition 2

The set of all integers $ℤ$ is not a field.

Proof: Proposition 2.

Let

𝑎 = 2 \in ℤ

. Then there is no multiplicative inverse of

𝑎

ℤ

, since there is no integer

𝑎^{- 1}

such that

𝑎 \cdot 𝑎^{- 1} = 2 𝑎^{- 1} = 1

. Thus,

ℤ

does not satisfy the field axiom of existence of multiplicative inverses, so

ℤ

is not a field. ∎

Proposition 3: Additional Properties of Fields

For the following, let $𝐹$ be a field with additive identity $0_{𝐹}$ .

Distributivity over Multiplication. $\forall 𝑎, 𝑏, 𝑐 \in 𝐹, (𝑎 + 𝑏) 𝑐 = 𝑎 𝑐 + 𝑏 𝑐$ .
Multiplication by Zero. $\forall 𝑎 \in 𝐹, 𝑎 \cdot 0_{𝐹} = 0_{𝐹}$ .
Double Negative Property. $\forall 𝑎 \in 𝐹, - (- 𝑎) = 𝑎$ .
Product of Negatives. $\forall 𝑎, 𝑏 \in 𝐹, (- 𝑎) (- 𝑏) = 𝑎 𝑏$ .
No Zero Divisors. $\forall 𝑎, 𝑏 \in 𝐹, if 𝑎 𝑏 = 0_{𝐹} then 𝑎 = 0_{𝐹} or 𝑏 = 0_{𝐹}$ .

Subfields

Definition 5: Subfield

Let $𝐹$ be a field, and let $𝐻$ be a subset of $𝐹$ . Then $𝐻$ is a subfield of $𝐹$ iff $𝐻$ is itself a field under the same operations of addition and multiplication as $𝐹$ .

Theorem 2: Subfield Criteria

Let $𝐹$ be a field, and let $𝐻$ be a subset of $𝐹$ . Let $0_{𝐹}$ and $1_{𝐹}$ denote the additive and multiplicative identities of $𝐹$ , respectively. Then $𝐻$ is a subfield of $𝐹$ if and only if the following criteria are met:

Existence of identities. $0_{𝐹} \in 𝐻$ and $1_{𝐹} \in 𝐻$ .
Closure under subtraction. $\forall 𝑎, 𝑏 \in 𝐻, 𝑎 - 𝑏 \in 𝐻$ .
Closure under division. $\forall 𝑎, 𝑏 \in 𝐻, 𝑏 \neq 0 \Rightarrow 𝑎 / 𝑏 \in 𝐻 .$

Proof: Subfield Criteria.

Suppose $𝐻$ is a subset of $𝐹$ . Let $𝑝$ be the property that $𝐻$ is a subfield of $𝐹$ , and let $𝑞$ be the property that $𝐻$ satisfies the three conditions listed in Theorem 2.

$𝒑 \Rightarrow 𝒒$ . Assume $𝐻$ is a subfield of $𝐹$ . Then $𝐻$ is a field under the same operations as $𝐹$ .

Existence of identities. By the definition of a field, $𝐻$ contains the additive and multiplicative identities of $𝐹$ , i.e. $0_{𝐹} \in 𝐻$ and $1_{𝐹} \in 𝐻$ .
Closure under subtraction. By the definition of a field, $𝐻$ must contain additive inverses, such that for any $𝑏 \in 𝐻$ , there must be a $- 𝑏 \in 𝐻$ . Since $𝐻$ is closed under addition, for any $𝑎 \in 𝐻$ , $𝑎 + (- 𝑏) = 𝑎 - 𝑏 \in 𝐻$ .
Closure under division. By the definition of a field, $𝐻$ must contain multiplicative inverses, such that for any $𝑏 \in 𝐻$ with $𝑏 \neq 0$ , there must be a $𝑏^{- 1} \in 𝐻$ . Since $𝐻$ is closed under multiplication, for any $𝑎 \in 𝐻$ , $𝑎 \cdot 𝑏^{- 1} = 𝑎 / 𝑏 \in 𝐻$ .

$𝒒 \Rightarrow 𝒑$ . Assume that $𝐻$ satisfies the three conditions listed in Theorem 2. Since $𝐻$ is a subset of $𝐹$ , the operations of addition and multiplication on $𝐻$ are inherited from $𝐹$ .

Associativity of addition and multiplication. Since $𝐻$ is a subset of $𝐹$ and the operations on $𝐻$ are inherited from $𝐹$ , the associativity of addition and multiplication in $𝐹$ implies the associativity of addition and multiplication in $𝐻$ .

Since $𝑝 \Rightarrow 𝑞$ and $𝑞 \Rightarrow 𝑝$ , we have $𝑝 \Leftrightarrow 𝑞$ . ∎

Exercises

Exercise 1.

Show that the set of rational numbers

ℚ

is a subfield of

ℝ

Exercise 2.

Show that the Boolean algebra

{0, 1}

is a field under Boolean addition (XOR) and Boolean multiplication (AND).

Exercise 3.

Prove Proposition 1.

Exercise 4.

Prove that Additive and Multiplicative Inverses are Unique.

Exercise 5.

Prove the Additional Properties of Fields.

Exercise 6.

Let

𝐹

be a field, and let

𝐻_{1}, 𝐻_{2}, \dots, 𝐻_{𝑛}

be a finite collection of subfields of

𝐹

. Show that the intersection of

𝐻_{1} \cap 𝐻_{2} \cap \dots \cap 𝐻_{𝑛}

is also a subfield of

𝐹

1.2 Vector Spaces

A vector is an element of a vector space, which is a fundamental structure in linear algebra. In fact, the study of vector spaces is what we call linear algebra.

Definition 6: Vector Space

A vector space $(𝑉, +, \cdot)$ over a field $𝐹$ , sometimes called an $𝑭$ -vector space, is a set $𝑉$ together with a binary operation $+$ (called vector addition) and a function $\cdot : 𝐹 \times 𝑉 \to 𝑉$ (called scalar multiplication) such that the following properties hold:

For all $𝑎, 𝑏 \in 𝐹$ and $𝐮, 𝐯, 𝐰 \in 𝑉$ :

Associativity of vector addition. $(𝐮 + 𝐯) + 𝐰 = 𝐮 + (𝐯 + 𝐰) .$
Commutativity of vector addition. $𝐮 + 𝐯 = 𝐯 + 𝐮 .$
Existence of vector additive identity. $\exists 𝟎 \in 𝑉 s.t. 𝐯 + 𝟎 = 𝐯 .$
Existence of vector additive inverses. $\exists (- 𝐯) \in 𝑉 s.t. 𝐯 + (- 𝐯) = 𝟎 .$
Compatibility of field multiplicative identity. $1_{𝐹} 𝐯 = 𝐯 .$
Distributivity of scalar multiplication over vector addition. $𝑎 (𝐮 + 𝐯) = 𝑎 𝐮 + 𝑎 𝐯 .$
Distributivity of scalar multiplication over scalar addition. $(𝑎 + 𝑏) 𝐯 = 𝑎 𝐯 + 𝑏 𝐯 .$
Compatibility of scalar and field multiplication. $(𝑎 𝑏) 𝐯 = 𝑎 (𝑏 𝐯) .$

Together, these are called the vector space axioms. The elements of $𝑉$ are called vectors. For any $𝐯 \in 𝑉$ , the element $- 𝐯$ is called the additive inverse of $𝐯$ .

Notation

Vectors are often denoted in boldface ( $𝐯$ ) or with an arrow on top ( $\vec{𝑣}$ ) to distinguish them from scalars.

Notation

A vector space $(𝑉, +, \cdot)$ over a field $𝐹$ is usually denoted simply as $𝑉$ , where the operations of vector addition and scalar multiplication are implied. When there is ambiguity, the notation $+_{𝑉}$ and $\cdot_{𝑉}$ may be used to refer to the vector addition and scalar multiplication operations of $𝑉$ , respectively.

Important

The definition of scalar multiplication implies that scalar multiplication is closed over $𝑉$ , meaning that for any scalar $𝑎 \in 𝐹$ and any vector $𝐯 \in 𝑉$ , the result of scalar multiplication $𝑎 𝐯$ is also an element of $𝑉$ .

Notation

The vector additive identity of a vector space $𝑉$ may be denoted $𝟎_{𝑉}$ when there is ambiguity.

Theorem 3: Multiplication of a vector by the zero scalar

Let $𝑉$ be a vector space over a field $𝐹$ . Then for any vector $𝐯 \in 𝑉$ , $0_{𝐹} 𝐯 = 𝟎_{𝑉}$ .

Proof: Theorem 3.

Let $𝑉$ be a vector space over a field $𝐹$ , and let $𝐯 \in 𝑉$ be an arbitrary vector. Call the field additive identity $0_{𝐹} \in 𝐹$ . Then:

\begin{matrix} 0_{𝐹} 𝐯 & = (0_{𝐹} + 0_{𝐹}) 𝐯 & by the definition of additive identity \\ 0_{𝐹} 𝐯 & = 0_{𝐹} 𝐯 + 0_{𝐹} 𝐯 & by distributivity of scalar multiplication over +_{𝐹} \\ 0_{𝐹} 𝐯 + (- 0_{𝐹} 𝐯) & = 0_{𝐹} 𝐯 + 0_{𝐹} 𝐯 + (- 0_{𝐹} 𝐯) & by left-adding - 0_{𝐹} 𝐯 to both sides \\ 𝟎_{𝑉} & = 0_{𝐹} 𝐯 + 0_{𝐹} 𝐯 + (- 0_{𝐹} 𝐯) & by the definition of vector additive inverse \\ 𝟎_{𝑉} & = 0_{𝐹} 𝐯 + (0_{𝐹} 𝐯 + (- 0_{𝐹} 𝐯)) & by associativity of vector addition \\ 𝟎_{𝑉} & = 0_{𝐹} 𝐯 + 𝟎_{𝑉} & by the definition of vector additive inverse \\ 𝟎_{𝑉} & = 0_{𝐹} 𝐯 & by the definition of vector additive identity & ∎ \end{matrix}

Theorem 4: Multiplication of the zero vector by a scalar

Let $𝑉$ be a vector space over a field $𝐹$ . Then for any scalar $𝑎 \in 𝐹$ , $𝑎 𝟎_{𝑉} = 𝟎_{𝑉}$ .

Proof: Theorem 4.

Let $𝑉$ be a vector space over a field $𝐹$ , and let $𝑎 \in 𝐹$ be an arbitrary scalar. Call the vector additive identity $𝟎_{𝑉} \in 𝑉$ . Then:

\begin{matrix} 𝑎 𝟎_{𝑉} & = 𝑎 (𝟎_{𝑉} + 𝟎_{𝑉}) & by the definition of vector additive identity \\ 𝑎 𝟎_{𝑉} & = 𝑎 𝟎_{𝑉} + 𝑎 𝟎_{𝑉} & by distributivity of scalar multiplication over +_{𝑉} \\ 𝑎 𝟎_{𝑉} + (- 𝑎 𝟎_{𝑉}) & = 𝑎 𝟎_{𝑉} + 𝑎 𝟎_{𝑉} + (- 𝑎 𝟎_{𝑉}) & by left-adding - 𝑎 𝟎_{𝑉} to both sides \\ 𝟎_{𝑉} & = 𝑎 𝟎_{𝑉} + 𝑎 𝟎_{𝑉} + (- 𝑎 𝟎_{𝑉}) & by the definition of vector additive inverse \\ 𝟎_{𝑉} & = 𝑎 𝟎_{𝑉} + (𝑎 𝟎_{𝑉} + (- 𝑎 𝟎_{𝑉})) & by associativity of vector addition \\ 𝟎_{𝑉} & = 𝑎 𝟎_{𝑉} + 𝟎_{𝑉} & by the definition of vector additive inverse \\ 𝟎_{𝑉} & = 𝑎 𝟎_{𝑉} & by the definition of vector additive identity & ∎ \end{matrix}

Property: Negation is Scalar Multiplication by $- 1$

Let $𝑉$ be a vector space over a field $𝐹$ . Then for any vector $𝐯 \in 𝑉$ , $(- 1) 𝐯 = - 𝐯$ .

Proof: Negation is Scalar Multiplication by

- 1

Let $𝑉$ be a vector space over a field $𝐹$ , and let $𝐯 \in 𝑉$ be an arbitrary vector. Call the field additive identity $0_{𝐹} \in 𝐹$ and the field multiplicative identity $1 \in 𝐹$ . By the definition of the additive inverse, there exists a scalar $- 1 \in 𝐹$ such that $1 + (- 1) = 0_{𝐹}$ . Then:

\begin{matrix} (1 + (- 1)) 𝐯 & = 0_{𝐹} 𝐯 & by right-multiplying both sides by 𝐯 \\ 1 𝐯 + (- 1) 𝐯 & = 0_{𝐹} 𝐯 & by distributivity of scalar multiplication over +_{𝐹} \\ 𝐯 + (- 1) 𝐯 & = 0_{𝐹} 𝐯 & by the definition of multiplicative identity \\ 𝐯 + (- 1) 𝐯 & = 𝟎_{𝑉} & by Theorem 3 \\ (- 𝐯) + 𝐯 + (- 1) 𝐯 & = 𝟎_{𝑉} + (- 𝐯) & by adding - 𝐯 to both sides \\ (- 𝐯) + 𝐯 + (- 1) 𝐯 & = - 𝐯 & by the definition of vector additive identity \\ 𝐯 + (- 𝐯) + (- 1) 𝐯 & = - 𝐯 & by commutativity of vector addition \\ 𝟎_{𝑉} + (- 1) 𝐯 & = - 𝐯 & by the definition of vector additive inverse \\ (- 1) 𝐯 & = - 𝐯 & by the definition of vector additive identity & ∎ \end{matrix}

The cartesian product of a field $𝐹$ with itself $𝑛$ times, denoted $𝐹^{𝑛}$ , is the set of all $𝑛$ -tuples of elements of $𝐹$ .

It turns out that for any field $𝐹$ and positive integer $𝑛$ , $𝐹^{𝑛}$ is a vector space over $𝐹$ under componentwise addition and scalar multiplication.

Notation

In the context of introducing $𝐹^{𝑛}$ , assume $𝑛$ is a positive integer.

Definition 7: The set $𝐹^{𝑛}$

Let $𝐹$ be a field. The set $𝐹^{𝑛}$ is the set of all $𝑛$ -tuples of elements of $𝐹$ :

𝐹^{𝑛} = {(𝑎_{1}, 𝑎_{2}, \dots, 𝑎_{𝑛}) | 𝑎_{1}, 𝑎_{2}, \dots, 𝑎_{𝑛} \in 𝐹} .

Definition 8: Operations on $𝐹^{𝑛}$

Define componentwise addition and scalar multiplication on $𝐹^{𝑛}$ as follows:

Componentwise addition. For any $𝐮 = (𝑢_{1}, 𝑢_{2}, \dots, 𝑢_{𝑛}), 𝐯 = (𝑣_{1}, 𝑣_{2}, \dots, 𝑣_{𝑛}) \in 𝐹^{𝑛}$ ,
$𝐮 + 𝐯 = (𝑢_{1} + 𝑣_{1}, 𝑢_{2} + 𝑣_{2}, \dots, 𝑢_{𝑛} + 𝑣_{𝑛}) .$
Scalar multiplication. For any scalar $𝑎 \in 𝐹$ and any vector $𝐯 = (𝑣_{1}, 𝑣_{2}, \dots, 𝑣_{𝑛}) \in 𝐹^{𝑛}$ ,
$𝑎 𝐯 = (𝑎 𝑣_{1}, 𝑎 𝑣_{2}, \dots, 𝑎 𝑣_{𝑛}) .$

Theorem 5: The set $𝐹^{𝑛}$ is a vector space over $𝐹$

Let $𝐹$ be a field. Then $𝐹^{𝑛}$ is a vector space over $𝐹$ under the operations defined in Definition 8.

Proof: Theorem 5.

Let

𝐹

be a field, and let

𝐹^{𝑛}

be the set as defined in Definition 7. Define vector addition and scalar multiplication on

𝐹^{𝑛}

as in Definition 8.

Example.

An element of

𝐹^{3}

is a triple

(𝑎, 𝑏, 𝑐)

where

𝑎, 𝑏, 𝑐 \in 𝐹

Example.

Elements of

ℝ^{2}

, a vector space over

ℝ

, can represent points in the 2D Cartesian plane.

Definition 9: Real and Complex Vector Spaces

A vector space over $ℝ$ is called a real vector space, and a vector space over $ℂ$ is called a complex vector space.

Corollary

$ℝ^{𝑛}$ is a real vector space, and $ℂ^{𝑛}$ is a complex vector space.

Subspaces

A subspace of an $𝐹$ -vector space $𝑉$ is a subset of $𝑉$ which is itself a vector space under the same vector addition and scalar multiplication as $𝑉$ .

Definition 10: Subspace

Let $𝑉$ be a vector space over a field $𝐹$ , and let $𝐻$ be a subset of $𝑉$ . Let $+_{𝑉}$ denote the vector addition operation on $𝑉$ , and let $\cdot_{𝑉}$ denote the scalar multiplication function on $𝑉$ using scalars from $𝐹$ .

Then $𝐻$ is a subspace of $𝑉$ iff $𝐻$ is itself a vector space under the vector addition $+_{𝑉}$ and scalar multiplication $\cdot_{𝑉}$ .

Theorem 6: Subspace Criteria

Let $𝑉$ be a vector space over a field $𝐹$ , and let $𝐻$ be a subset of $𝑉$ . Let $𝟎_{𝑉}$ denote the vector additive identity of $𝑉$ . Then $𝐻$ is a subspace of $𝑉$ if and only if the following criteria are met:

Existence of additive identity. $𝟎_{𝑉} \in 𝐻$ .
Closure under vector addition. $\forall 𝐮, 𝐯 \in 𝐻, 𝐮 + 𝐯 \in 𝐻 .$
Closure under scalar multiplication. $\forall 𝑎 \in 𝐹, \forall 𝐯 \in 𝐻, 𝑎 𝐯 \in 𝐻 .$

Proof: Subspace Criteria.

Suppose $𝐻$ is a subset of $𝑉$ . Let $𝑝$ be the property that $𝐻$ is a subspace of $𝑉$ , and let $𝑞$ be the property that $𝐻$ satisfies the three conditions listed in Theorem 6.

$𝒑 \Rightarrow 𝒒$ . Assume $𝐻$ is a subspace of $𝑉$ . TODO

$𝒒 \Rightarrow 𝒑$ . Assume that $𝐻$ is non-empty, closed under vector addition, and closed under scalar multiplication. TODO

Since $𝑝 \Rightarrow 𝑞$ and $𝑞 \Rightarrow 𝑝$ , we have $𝑝 \Leftrightarrow 𝑞$ . ∎

Property: ${𝟎}$ is a subspace of every vector space

If $𝟎_{𝑉}$ is the vector additive identity of a vector space $𝑉$ , then ${𝟎_{𝑉}}$ is a subspace of $𝑉$ .

Proof:

{𝟎}

is a subspace of every vector space.

Suppose $𝑉$ is a vector space over $𝐹$ , and $𝟎_{𝑉}$ is the vector additive identity of $𝑉$ . Since $𝟎_{𝑉} \in 𝑉$ , it follows that ${𝟎_{𝑉}} \subseteq 𝑉$ .

Existence of additive identity. $𝟎_{𝑉} \in {𝟎_{𝑉}} .$
Closure under vector addition. For any $𝐮, 𝐯 \in {𝟎_{𝑉}}$ , we have $𝐮 = 𝐯 = 𝟎_{𝑉}$ , so $𝐮 + 𝐯 = 𝟎_{𝑉} + 𝟎_{𝑉} = 𝟎_{𝑉} \in {𝟎_{𝑉}}$ .
Closure under scalar multiplication. For any scalar $𝑎 \in 𝐹$ and any vector $𝐯 \in {𝟎_{𝑉}}$ , we have $𝐯 = 𝟎_{𝑉}$ , so $𝑎 𝐯 = 𝑎 𝟎_{𝑉} = 𝟎_{𝑉} \in {𝟎_{𝑉}}$ .

By Theorem 6, ${𝟎_{𝑉}}$ is a subspace of $𝑉$ . ∎

Property: Every vector space is a subspace of itself

If $𝑉$ is a vector space, then $𝑉$ is a subspace of itself, $𝑉$ .

Exercises

Exercise 7.

Show that

ℝ

is a vector space over

ℚ

, the field of rational numbers.

Exercise 8.

Prove the property that Every vector space is a subspace of itself.

Exercise 9.

Let

𝐹

be a field.

𝐹^{\infty}

is the set of all infinite sequences of elements of

𝐹

, that is,

𝐹^{\infty} = {(𝑎_{1}, 𝑎_{2}, 𝑎_{3}, \dots) | 𝑎_{1}, 𝑎_{2}, 𝑎_{3}, \dots \in 𝐹} .

Show that

𝐹^{\infty}

is a vector space over

𝐹

under componentwise addition and scalar multiplication.

Exercise 10.

Show that the set of all polynomials with real coefficients,

ℙ (ℝ)

, is a vector space over

ℝ

under polynomial addition and scalar multiplication.

Exercise 11.

Let

𝐶_{[0, 1]} (ℝ)

be the set of all real-valued functions that are continuous on the closed interval

[0, 1]

. Show that

𝐶_{[0, 1]} (ℝ)

is a vector space over

ℝ

Exercise 12.

Recall $𝐶_{[0, 1]} (ℝ)$ from Exercise 11. Let $𝐻$ be a subset of $𝐶_{[0, 1]} (ℝ)$ defined by:

𝐻 = {𝑓 \in 𝐶_{[0, 1]} (ℝ) | \int_{0}^{1} 𝑓 (𝑥) d 𝑥 = 0} .

Show that $𝐻$ is a subspace of $𝐶_{[0, 1]} (ℝ)$ .

Exercise 13.

Let

𝑉

be a vector space, and let

𝐻_{1}, 𝐻_{2}, \dots, 𝐻_{𝑝}

be subspaces of

𝑉

. Show that the intersection

𝐻_{1} \cap 𝐻_{2} \cap \dots \cap 𝐻_{𝑝}

is also a subspace of

𝑉

Exercise 14.

Let

𝐻_{1}, 𝐻_{2}

be subspaces of a vector space

𝑉

. Show that

𝐻_{1} \cup 𝐻_{2}

is a subspace if and only if

𝐻_{1} \subseteq 𝐻_{2}

𝐻_{2} \subseteq 𝐻_{1}

typst test 2

Fundamental Structures

1.1 Fields

Definition 1: Binary Operation

Definition 2: Field

Notation

Notation

Notation

Property: Additive and Multiplicative Identities are Unique

Property: Additive and Multiplicative Inverses are Unique

Definition 3: Field Subtraction

Definition 4: Field Division

Notation

Examples and Properties of Fields

Theorem 1

Proposition 1

Proposition 2

Proposition 3: Additional Properties of Fields

Subfields

Definition 5: Subfield

Theorem 2: Subfield Criteria

Exercises

1.2 Vector Spaces

Definition 6: Vector Space

Notation

Notation

Important

Notation

Theorem 3: Multiplication of a vector by the zero scalar

Theorem 4: Multiplication of the zero vector by a scalar

Property: Negation is Scalar Multiplication by −1

Notation

Definition 7: The set 𝐹𝑛

Definition 8: Operations on 𝐹𝑛

Theorem 5: The set 𝐹𝑛 is a vector space over 𝐹

Definition 9: Real and Complex Vector Spaces

Corollary

Subspaces

Definition 10: Subspace

Theorem 6: Subspace Criteria

Property: {𝟎} is a subspace of every vector space

Property: Every vector space is a subspace of itself

Exercises

Property: Negation is Scalar Multiplication by $- 1$

Definition 7: The set $𝐹^{𝑛}$

Definition 8: Operations on $𝐹^{𝑛}$

Theorem 5: The set $𝐹^{𝑛}$ is a vector space over $𝐹$

Property: ${𝟎}$ is a subspace of every vector space