Vector Spaces

Overview

Let $F$ be a field. A vector space over $F$ is an abelian group $V$ under addition together with an operation of scalar multiplication of each element of $V$ by each element of $F$ on the left, such that for all $a, b \in F$ and $v, w \in V$ the following conditions hold:

Closure under scalar multiplication. $a v \in V$ ;
Associativity of scalar multiplication. $a (b v) = (a b) v$ ;
Scalar multiplication identity. $1 v = v$ ;
Distributivity I. $(a + b) v = a v + b v$ ;
Distributivity II. $a (v + w) = a v + a w$ .

Let $V$ be a vector space. The elements of $V$ are vectors and the elements of $F$ are scalars.

Subspaces

Let $V$ be a vector space over field $F$ . If $U \subseteq V$ is a vector space over $F$ under the same addition and scalar multiplication as $V$ , then $U$ is a subspace of $V$ .

Criterion

A nonempty subset $U$ of a vector space $V$ is a subspace if and only if

$U$ is closed under addition, and
$U$ is closed under scalar multiplication.

Sums

Let $V$ be a vector space over a field $F$ and $U_{1}, U_{2}, \dots, U_{n}$ be subspaces of $V$ . Then their sumset (or simply sum) is the smallest subspace of $V$ containing each of $U_{1}, U_{2}, \dots, U_{n}$ , denoted as

\sum_{i = 1}^{n} U_{i} = U_{1} + U_{2} + \dots + U_{n} .

Unions

Let $V$ be a vector space and $U, W \leq V$ . Then $U \cup W$ is a subspace if and only if either $U \subseteq W$ or $W \subseteq U$ .

Intersections

Let ${U_{i} ∣ i \in I}$ be an indexed set of subspaces of vector space $V$ . Then $⋂_{i \in I} U_{i}$ is a subspace of $V$ .

Direct Products

Let $V_{1}, V_{2}, \dots, V_{n}$ be vector spaces over a common field $F$ . Let $λ \in F$ and $a_{i}, b_{i} \in V_{i}$ for $1 \leq i \leq n$ . Then external direct product $⨂_{i = 1}^{n} V_{i}$ is a vector space over $F$ under operations

\begin{aligned} (a_{1}, a_{2}, \dots, a_{n}) + (b_{1}, b_{2}, \dots, b_{n}) & = (a_{1} + b_{1}, a_{2} + b_{2}, \dots, a_{n} + b_{n}) \\ λ (a_{1}, a_{2}, \dots, a_{n}) & = (λ a_{1}, λ a_{2}, \dots, λ a_{n}) \end{aligned}

Direct Sums

Let $V$ be a vector space over field $F$ and $U_{1}, U_{2}, \dots, U_{n} \leq V$ . Then sumset $\sum U_{i}$ is an internal direct sum if each element of $\sum U_{i}$ can be written in only one way as a sum $u_{1} + \dots + u_{m}$ , where each $u_{i} \in U_{i}$ .

This holds if and only if $0 \in \sum U_{i}$ is uniquely written as $0 + 0 + \dots + 0$ .

Linear Maps

Let $V$ and $W$ be vector spaces over field $F$ . A linear map is a homomorphism between vector spaces. That is, a linear map $ϕ : V \to W$ is a function that satisfies the following conditions for all $a, b \in V$ and $λ \in F$ :

Additivity. $ϕ (a + b) = ϕ (a) +^{'} ϕ (b)$
Homogeneity. $ϕ (λ a) = λ ϕ (a)$

Note that $ϕ$ is also a group homomorphism so the linked structural properties are preserved. Additionally,

Linear Independence
- If $ϕ (v_{1}), \dots, ϕ (v_{n})$ is linearly independent, then so is $v_{1}, \dots, v_{n}$ .
- If $ϕ$ is injective and $v_{1}, \dots, v_{n}$ is linearly independent, then so is $ϕ (v_{1}), \dots, ϕ (v_{n})$ .
Subspaces
- If $U \leq V$ , then $ϕ [[U]] \leq W$ .
- If $U \leq W$ , then $ϕ^{- 1} [[U]] \leq V$ .

Restrictions

Let $T \in L (V, W)$ and $U \leq V$ . Then $T ↾ U$ is a linear map from $U$ to $W$ .

Images

Let $T \in L (V, W)$ . Then the image of $T$ , denoted $Im (T)$ , is a subspace of $W$ consisting of those vectors of form $T v$ for some $v \in V$ . That is,

Im (T) = {T v ∣ v \in V} .

The dimension of the image is the rank of $T$ .

Kernels

Let $T \in L (V, W)$ . Then the kernel of $T$ , denoted $Ker (T)$ , is a subspace of $V$ consisting of those vectors that $T$ maps to $0$ . That is,

Ker (T) = {v \in V ∣ T v = 0} .

$T$ is injective if and only if $Ker T = {0}$ . The dimension of the kernel is the nullity of $T$ .

Invertibility

Let $V$ and $W$ be vector spaces. A linear map $T \in L (V, W)$ is invertible (or bijective) if there exists a linear map $S \in L (W, V)$ such that $S T$ equals the identity map on $V$ and $T S$ equals the identity map on $W$ . Such an $S$ is a (multiplicative) inverse of $T$ .

Let $U$ , $V$ , and $W$ be any vector spaces. Let $S \in L (V, W)$ and $T \in L (U, V)$ . If $S T$ is invertible, then $S$ is surjective and $T$ is injective. If $U$ and $V$ are finite-dimensional with $dim U = dim V$ , then $T$ is injective if and only if $T$ is surjective.

Isomorphisms

An isomorphism is an invertible linear map. Two vector spaces are isomorphic if there is an isomorphism from one vector space onto the other. Two finite-dimensional vector spaces are isomorphic if and only if they have the same dimension.

Linear Map Space

Let $V$ and $W$ be vector spaces over field $F$ . The set of linear maps from $V$ to $W$ is denoted $L (V, W)$ . This set is a vector space under the following definitions of addition and scalar multiplication:

\begin{aligned} (S + T) (v) & = S v + T v \\ (λ S) (v) & = λ (S v) \end{aligned}

If $T \in L (U, V)$ and $S \in L (V, W)$ , we define the product $S T \in L (U, W)$ as $(S T) (v) = S (T v)$ . That is, the product is normal function composition. The product operation satisfies:

Associativity. $(T_{1} T_{2}) T_{3} = T_{1} (T_{2} T_{3})$
Identity. $T I = I T = T$ where $I$ is the identity map.
Distributivity. $(S_{1} + S_{2}) T = S_{1} T + S_{2} T$ and $S (T_{1} + T_{2}) = S T_{1} + S T_{2}$ .

A linear map from a vector space to itself is called an operator. The notation $L (V)$ denotes the set of all operators on $V$ . That is, $L (V) = L (V, V)$ .

Special Cases

Functions

Let $F$ be a field and $S$ be a nonempty set. Then $F^{S}$ , the set of all functions from $S$ to $F$ , is a vector space under the following operations of addition and scalar multiplication:

Addition. For $f, g \in F^{S}$ , $f + g \in F^{S}$ is given by $(f + g) (x) = f (x) + g (x)$ .
Scalar multiplication. For $λ \in F$ and $f \in F^{S}$ , $λ f \in F^{S}$ is given by $(λ f) (x) = λ f (x)$ .

Polynomials

Let $F$ be a field. Then $P (F)$ , the set of all polynomials with coefficients in $F$ , is a vector space under the usual operations of addition and scalar multiplication. Similarly, for any $n \in N$ , $P_{n} (F)$ is also a vector space.

$P_{n} (F)$ is a subspace of $P (F)$ which is a subspace of $F^{F}$ .

Real Coordinates

The standard basis vectors in $R^{2}$ are defined as $i = ⟨ 1, 0 ⟩$ and $j = ⟨ 0, 1 ⟩$ . Those in $R^{3}$ are defined as $i = ⟨ 1, 0, 0 ⟩$ , $j = ⟨ 0, 1, 0 ⟩$ , and $k = ⟨ 0, 0, 1 ⟩$ . This is naturally generalized to vector space $R^{n}$ for any positive integer $n$ .

A vector with initial point at the origin is said to be in standard position.