Vector Spaces

In Vectors and Matrices we have briefly addressed the idea of Vector Space. Vector space is like an abstraction of linear systems by only relying on axioms for definition and develop the theory base on that. In this way, any systems that match the axioms are considered as vector space and the theories can then be applied. Linear algebra can then be summarised as the study of vector spaces and linear maps between them.

Definition. Let $\mathbb{F}$ be an arbitrary field. A vector space $V$ over $\mathbb{F}$ or $\mathbb{F}$-vecor space is a non-empty set of vectors, a field $\mathbb{F}$ of scalars, together with two binary operations: an addition $a + b$ of vectors $a$ and $b$ and a scalar multiplication $\lambda v$ of a scalar $\lambda$ and a vector $v$. The operations are assumed to satisfy the following axioms:

  • $V$ is closed under addition, i.e.

    \[(\forall a, b \in V)\; a + b \in V\]

  • addition is commutative, i.e. for all $a, b \in V$

    \[a + b = b + a\]

  • addition is associative, i.e. for all $a, b, c \in V$

    \[a + (b + c) = (a + b) + c\]

  • addition has an identity element, i.e. there exists an element $0 \in V$ such that

    \[a + 0 = a\]

  • for all $a \in V$, there exists an additive inverse $a’ \in V$ such that

    \[a + a' = 0\]

  • $V$ is closed under scalar multiplication, i.e.

    \[(\forall \lambda \in F)(\forall a \in V)\; \lambda a \in V\]

  • scalar multiplication is distributive over scalar addition, i.e. for all $\lambda, \mu \in \mathbb{F}$ and $a \in V$

    \[(\lambda + \mu)a = \lambda a + \mu a\]

  • scalar multiplication is distributive over vector addition, i.e. for all $\lambda \in \mathbb{F}$ and $a, b \in V$

    \[\lambda (a + b)= \lambda a + \lambda b\]

  • scalar multiplication is “associative”, i.e. for all $\lambda, \mu \in \mathbb{F}$ and $a \in V$

    \[\lambda (\mu a) = (\lambda \mu) a\]

  • scalar multiplication has an identity element, i.e. there exists an element $1 \in \mathbb{F}$ such that

    \[1 a = a\]

Note that $(V, +)$ is an abelian group and therefore the additive identity and inverse are unique. Some of the other properties are proved in Vectors and Matrices.

Subspaces

Definition. A nonempty subset $U \subset V$ is a subspace if

  • $U$ is closed under addition, i.e.

    \[(\forall u_1, u_2 \in U)\; u_1 + u_2 \in U\]

  • $U$ is closed under scalar multiplication, i.e.

    \[(\forall \lambda \in F)(\forall u \in U)\; \lambda u \in U\]

In short, $U \subset V$ is a subspace iff $U \not= \emptyset$ and

\[(\forall u_1, u_2 \in U)(\lambda, \mu \in \mathbb{F})\; \lambda u_1 + \mu u_2 \in U\]

Proposition. Suppose that $U$ and $W$ are subspaces of a vector space $V$. Then $U \cap W$ is also a subspace of $V$.

Proof.

$U \cap W$ contains $0$ so it is nonempty. For all $v_1, v_2 \in U \cap W$, $\lambda v_1 + \mu v_2 \in U$ and $\lambda v_1 + \mu v_2 \in W$ so $\lambda v_1 + \mu v_2 \in U \cap W$.

Quotient Spaces

Definition. Suppose that $V$ is a vector space over $\mathbb{F}$ and $U$ is a subspace of $V$. Then the quotient space $V/U$ is the set $\Set{v + U}$ with addition

\[(v_1 + U) + (v_2 + U) = (v_1 + v_2) + U\]

and scalar multiplication

\[\lambda (v + U) = (\lambda v) + U\]

It is easier to understand the above structure by considering the equivalence relation $v_1 \sim v_2$ if $v_1 - v_2 \in U$. Therefore, the equivalence class of $v$ is defined as

\[[v] = \Set{v + u : u \in U}\]

which is denoted as $v + U$ above and $V/U$ is the set containing all the equivalence classes induced by $\sim$ on $U$ with addition $[v_1] + [v_2] = [v_1 + v_2]$ and $\lambda [v] = [\lambda v]$. Note that all the elements in $U$ are in the equivalence class $[0]$ so the quotient space $V/U$ is obtained by “collapsing” $U$ to zero.

Proposition. The quotient space $V/U$ is a vector space over $\mathbb{F}$.

Proof.

For $v_1, v_1’, v_2, v_2’ \in V$ and $u_1, u_2 \in U$ with $[v_1] = [v_1’]$ and $[v_2] = [v_2’]$, we have

\[v_1' + v_2' = (v_1 + u_1) + (v_2 + u_2) = (v_1 + v_2) + (u_1 + u_2)\]

so $[v_1’ + v_2’] = [v_1 + v_2]$ since $u_1 + u_2 \in U$. Similarily,

\[\lambda v_1' = \lambda (v_1 + u_1) = (\lambda v_1) + (\lambda u_1)\]

so $[\lambda v_1’] = [\lambda v_1]$. Thus, both operations are well-defined, i.e. independent of the choice of representatives.

The axioms are an almost immediate consequence of the fact that the same axioms hold for $V$ and $U$ being a vector space, e.g.

\[\lambda (\mu [v]) = [\lambda (\mu v)] = [(\lambda \mu) v] = (\lambda \mu) [v]\]

Direct Sums

Definition. Suppose that $U$ and $W$ are subspaces of a vector space $V$ over $\mathbb{F}$. Then the sum of $U$ and $W$ is defined by the set

\[U + W = \Set{u + w : u \in U, w \in W}\]

Proposition. $U + W$ is a subspace of $V$.

Proof.

$U + W$ is nonempty since it contains $0$. For all $u_1 + w_1, u_2 + w_2 \in U + W$,

\[\lambda (u_1 + w_1) + \mu (u_2 + w_2) = (\lambda u_1 + \mu u_2) + (\lambda w_1 + \mu w_2) \in U + W\]

Definition. $V$ is the (internal) direct sum of $U$ and $W$, written $V = U \oplus W$, if every element $v \in V$ can be written uniquely as $u + w$ with $u \in U$ and $w \in W$. $U$ and $W$ are said to be the complementary subspaces in $V$.

Proposition. $V = U \oplus W$ iff $V = U + W$ and $U \cap W = \Set{0}$.

Proof.

Obviously, $V = U + W$ and if $v \in U \cap W \not= 0$, then $v = 0 + v = v + 0$ which cannot be written uniquely.

Definition. The (external) direct sum $U \oplus W$ of two vector spaces $U$ and $W$ over $\mathbb{F}$ is defined by the set

\[U \oplus W = \Set{(u, w) : u \in U, w \in W}\]

with the natural coordinate-wise operations.

Proposition. $U \oplus W$ is a vector space over $\mathbb{F}$ and is the internal direct sum of the subspaces

\[\Set{(u, 0) : u \in U} \quad \text{and} \quad \Set{(0, v) : v \in V}\]

The definition can be extended to $n$ vector spaces.

Reference