1 Vector Spaces

1.1 Fields

Field

A field $F$ satisfies the following axioms:

1. $\forall a, b \in F$, $a + b \in F$, $a \cdot b \in F$.
2. $\forall a, b, c \in F$, $(a + b) + c = a + (b + c)$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
3. $\forall a, b \in F$, $a + b = b + a$, $a \cdot b = b \cdot a$.
4. There exists an element $0 \in F$ such that $\forall a \in F$, $a + 0 = a$.
5. There exists an element $1 \in F$, with $1 \neq 0$, such that $\forall a \in F$, $a \cdot 1 = a$.
6. $\forall a \in F$, there exists an element $-a \in F$ such that $a + (-a) = 0$.
7. $\forall a \in F \setminus \{0\}$, there exists an element $a^{-1} \in F$ such that $a \cdot a^{-1} = 1$.
8. $\forall a, b, c \in F$, $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$.

Examples

1. $\mathbb{Q}$ — the field of rational numbers.

2. $\mathbb{R}$ — the field of real numbers.

3. $\mathbb{C}$ — the field of complex numbers.

4. $\mathbb{F}_p$ — the finite field with $p$ elements, where $p$ is a prime number.

1.2 Vector Spaces

Vector Space

A vector space $V$ over a field $F$ satisfies the following axioms:

1. $\forall x, y \in V$, $x + y \in V$.
2. $\forall a \in F, \forall x \in V$, $ax \in V$.
3. $\forall x, y \in V$, $x + y = y + x$.
4. $\forall x, y, z \in V$, $(x + y) + z = x + (y + z)$.
5. There exists an element $0 \in V$ such that $\forall x \in V$, $x + 0 = x$.
6. $\forall x \in V$, there exists an element $-x \in V$ such that $x + (-x) = 0$.
7. $\forall x \in V$, $1x = x$.
8. $\forall a,b \in F, \forall x \in V$, $(ab)x = a(bx)$.
9. $\forall a \in F, \forall x, y \in V$, $a(x + y) = ax + ay$.
10. $\forall a, b \in F, \forall x \in V$, $(a + b)x = ax + bx$.

• The elements of the field $F$ are called scalars and the elements of the vector space $V$ are called vectors.

Examples

1. $\text{F}_n$ — the set of all $n$-tuples with entries from a field $F$.

2. $\text{M}_{n \times m}(F)$ — the set of all $n \times m$ matrices with entries from a field $F$.

3. $\text{P}_n(F)$ — all polynomials with coefficients in $F$ of degree $\leq n$.

$$\begin{equation*} \text{P}_n(F) = \{\, a_0 + a_1x + \dots + a_nx^n \mid a_i \in F \,\} \end{equation*}$$

4. $\text{P}(F)$ — all polynomials with coefficients in $F$.

5. $\mathcal{F}(S, F)$ — Let $S$ be any nonempty set and $F$ be any field, and $\mathcal{F}(S, F)$ denote the set of all functions from $S$ to $F$.

1.3 Subspaces

Subspace

A subset $W$ of a vector space $V$ over a field $F$ is called a subspace of $V$ with the operations of addition and scalar multiplication defined on $V$ satisfies the following axioms:

1. $0 \in W$.
2. $\forall x, y \in W$, $x + y \in W$.
3. $\forall a \in F, \forall x \in W$, $ax \in W$.

Theorem 1.1. Any intersection of subspaces of a vector space $V$ is a subspace of $V$.

1.4 Linear Combinations

Linear Combination

Let $V$ be a vector space and $S$ be a nonempty subset of $V$. A vector $v \in V$ is called a linear combination of $S$. if there exist a finite number of vectors $u_1, u_2, \ldots, u_n \in S$ and scalars $a_1, a_2, \ldots, a_n \in F$ such that $v = a_1u_1 + a_2u_2 + \cdots + a_nu_n$.

• In this case, we also say that $v$ is a linear combination of $u_1, u_2, \ldots, u_n$ and call $a_1, a_2, \ldots, a_n$ the coefficients of the linear combination.

• The zero vector is a linear combination of any nonempty subset of (V).

Span

Let $S$ be a nonempty subset of a vector space $V$. The span of $S$, denoted $\text{span}(S)$, is the set consisting of all linear combinations of the vectors in $S$.

• $\text{span}(S)$ is a subspace of $V$.

• if $\text{span}(S) = V$, we also say $S$ generate (or span) $V$.

1.5 Linear Dependence and Linear Independence

Linearly Dependent and Linearly Independent

A subset $S$ of a vector space $V$ is called linearly dependent if there exist a finite number of distinct vectors $u_1, u_2, \ldots, u_n$ in $S$ and scalars $a_1, a_2, \ldots, a_n$, not all zero, such that

$$\begin{equation*} a_1u_1 + a_2u_2 + \cdots + a_nu_n = 0. \end{equation*}$$

A subset $S$ of a vector space $V$ is not linearly dependent is called linearly independent.

Theorem 1.2. Let $V$ be a vector space, and let $S_1 \subseteq S_2 \subseteq V$. If $S_2$ is linearly independent, then $S_1$ is linearly independent.

Theorem 1.3. Let $S$ be a linearly independent subset of a vector space $V$, and let $v$ be a vector in $V$ that is not in $S$. Then $S \cup \{v\}$ is linearly dependent if and only if $v \in \text{span}(S)$.

1.6 Bases and Dimension

Basis

A basis $\beta$ for a vector space $V$ is a linearly independent subset of $V$ that generates $V$.

Theorem 1.4. Let $V$ be a vector space and $\beta = \{u_1, u_2, \ldots, u_n\}$ be a subset of $V$. Then $\beta$ is a basis for $V$ if and only if each $v \in V$ can be uniquely expressed as a linear combination of vectors of $\beta$, that is, can be expressed in the form $v = a_1u_1 + a_2u_2 + \cdots + a_nu_n$ for unique scalars $a_1, a_2, \ldots, a_n$.

Theorem 1.5 (Replacement Theorem). Let $V$ be a vector space that is generated by a set $G$ containing exactly $n$ vectors, and let $L$ be a linearly independent subset of $V$ containing exactly $m$ vectors. Then $m \le n$ and there exists a subset $H$ of $G$ containing exactly $n - m$ vectors such that $L \cup H$ generates $V$.

Corollary 1. Let $V$ be a vector space having a finite basis. Then every basis for $V$ contains the same number of vectors.

Standard Basis

1. In $\text{F}^n$, let $e_1 = (1, 0, 0, \ldots, 0)$, $e_2 = (0, 1, 0, \ldots, 0)$, $\ldots$ ,$e_n = (0, 0, \ldots, 0, 1)$. $\{e_1, e_2, \ldots, e_n\}$ is a basis for $\text{F}^n$ and is called the standard basis for $\text{F}^n$.

2. In $\text{P}_n(\text{F})$ the set $\{1, x, x^2, \ldots, x^n\}$ is a basis. We call this basis the standard basis for $\text{P}_n(\text{F})$.

Dimension

A vector space is called finite-dimensional if it has a basis consisting of a finite number of vectors.

The unique number of vectors in each basis for $V$ is called the dimension of $V$ and is denoted by $\dim(V)$.

A vector space that is not finite-dimensional is called infinite-dimensional.

Examples

1. The vector space $\{0\}$ has dimension zero.
2. The vector space $\text{F}^n$ has dimension $n$.
3. The vector space $\text{M}_{n \times m}(\text{F})$ has dimension $nm$.
4. The vector space $\text{P}_n(\text{F})$ has dimension $n + 1$.

Theorem 1.6. Let $W$ be a subspace of a finite-dimensional vector space $V$. Then $W$ is finite-dimensional and $\dim(W) \le \dim(V)$. Moreover, if $\dim(W) = \dim(V)$, then $V = W$.

Theorem 1.7. If $W$ is a subspace of a finite-dimensional vector space $V$, then any basis for $W$ can be extended to a basis for $V$.