Subvector Spaces

Introduction

Abstract We introduce the concept of subvector space of $\RNrSpc{n}$. Then we present three ways of constructing subvector spaces: intersection, span, and orthogonal complement.

Outline Intuitively speaking, a subvector space, or shorter ‘subspace’, of $\RNrSpc{n}$ is a subset $V$ of $\RNrSpc{n}$ which is like a copy of $\RNrSpc{k}$ with $k\leq n$. For example, in $\RNrSpc{2}$ any line through the origin is a subvector space. In $\RNrSpc{3}$ any line or any plane through the origin is a subvector space. In general, we recognize a subvector space by these three properties: contains the $\Vect{0}$-vector, and is closed under vector addition and scalar multiplication.

Subvector spaces ‘arise naturally’. Below and in the following section we discuss three operations each of which yields a subvector space: orthogonal complement, intersection, and spanning.

Orthogonal complement is subvector space Given any collection $S$ of vectors in $\RNrSpc{n}$, its orthogonal complement $S^{\bot}$ is always a subspace of $\RNrSpc{n}$. The orthogonal complement of $S$ consists of all vectors those $\Vect{v}$ in $\RNrSpc{n}$ which are perpendicular to every $\Vect{s}$ in $S$. We encounter the orthogonal complement operation in various guises. For example, consider an $(m,n)$-matrix $A$.

The solutions of the homogeneous system of linear equations with unaugmented coefficient matrix $A$ consists of vectors whose dot product with every row of $\Mtrx{A}$ vanishes. Such vectors form exactly the orthogonal complement of the row vectors of $\Mtrx{A}$. So, it is a subspace of $\RNrSpc{n}$.
Equivalently, the solutions of the matrix equation $A\Vect{x}=\Vect{0}$ form the orthogonal complement of the row vectors of $\Mtrx{A}$. If looked at from this perspective, this subspace of $\RNrSpc{n}$ is called the null space of $\Mtrx{A}$.
Equivalently, consider the matrix transformation
\[L(\Vect{x}) = \Mtrx{A}\cdot \Vect{x}\]
The collection of vectors $\Vect{x}$ in $\RNrSpc{n}$ which get transformed into the $\Vect{0}$-vector of $\RNrSpc{m}$ forms the orthogonal complement of the row vectors of $\Mtrx{A}$. When viewed from this perspective, this subspace of $\RNrSpc{n}$ is called the kernel of $L$.

More generally, the kernel of any linear transformation is always a subspace of its domain.

Intersection of subspaces is a subspace Consider now two subspaces $V$ and $W$ of $\RNrSpc{n}$. From these we form its intersection $V \intrsctn W$, consisting of all those vectors which belong to both $V$ and $W$. Then we always find the $\Vect{0}$-vector in $V\intrsctn W$, and we show further that $V\intrsctn W$ is closed under vector addition and scalar multiplication. So it is a subspace of $\RNrSpc{n}$. By taking intersections repeatedly we see that the intersection $V_1\intrsctn \dots \intrsctn V_r$ of subspaces $V_1,\dots ,V_r$ is again a subspace of $\RNrSpc{n}$.

DefinitionSubvector space

A subvector space, or subspace, of $\RNrSpc{n}$ is a subset $V$ of $\RNrSpc{n}$ with the following properties

The $\Vect{0}$-vector belongs to $V$.
$V$ is closed under vector addition; that is, if $\Vect{x}$ and $\Vect{y}$ are in $V$, then $\Vect{x}+\Vect{y}$ is also in $V$.
$V$ is closed under scalar multiplication; that is, if $\Vect{x}$ is in $V$ and $t$ is a number in $\RNr$, then $t \Vect{x}$ is in $V$.

If the context is clear we may say ‘subspace’ instead of ‘subvector space’. Thus, if $V,W$ are subspaces of $\RNrSpc{n}$, we say that $V$ is a subspace of $W$ if $V$ is contained in $W$.

One way of generating subvector spaces is the orthogonal complement operation:

DefinitionOrthogonal complement

The orthogonal complement of a subset $S$ in a subvector space $V$ of $\RNrSpc{n}$ is

$S^{\bot}$

$\DefEq $

$\Set{ \Vect{x}\in V \st \DotPr{ \Vect{x} }{ \Vect{s} }=0,\ \ \text{for all $\Vect{s}\in S$} }$

The following proposition confirms that the orthogonal complement of $S$ is indeed a subspace.

PropositionOrthogonal complement is subspace

The orthogonal complement of a set $S$ in a subvector space $V$ of $\RNrSpc{n}$ is a subspace of $V$.

An example of the orthogonal complement operation is the null space of a matrix:

DefinitionNull space

The null space of an $(m,n)$-matrix $\Mtrx{A}$ is the collection of all those $\Vect{x}$ in $\RNrSpc{n}$ with $\Mtrx{A}\cdot \Vect{x} = \Vect{0}$. We denote it by $\NullSpc{ \Mtrx{A} }$

In the following proposition we identify the null space of a matrix $\Mtrx{A}$ as the orthogonal complement of the row vectors of $\Mtrx{A}$.

PropositionNull space and orthogonal complement

The null space of an $(m,n)$-matrix $\Mtrx{A}$ is the orthogonal complement of the row vectors of $\Mtrx{A}$ in $\RNrSpc{n}$. Therefore $\NullSpc{A}$ is a subvector space of $\RNrSpc{n}$.

DefinitionKernel of a linear map

Let $V$ and $W$ be subspaces of $\RNrSpc{n}$. The kernel of a linear transformation $L\from V\to W$ is

$\KerOf{L}$

$\DefEq $

$\Set{ \Vect{x}\in V \st L(\Vect{x}) = \Vect{0} }$

PropositionKernel is a subspace

If $V$ and $W$ are subvector spaces of $\RNrSpc{n}$, then the kernel of a linear map $L\from V\to W$ is a subspace of $V$. Moreover, if the $(m,n)$-matrix $\Mtrx{A}$ represents $L$, then $\KerOf{L} = \NullSpc{\Mtrx{A}}$

If we are given two subvector spaces, we can always generate a new one via their intersection:

DefinitionIntersection of subspaces

The intersection of subspaces $V$ and $W$ of $\RNrSpc{n}$ is denoted $V\cap W$ and consists of all those vectors in $\RNrSpc{n}$ which belong to both $V$ and $W$.

PropositionIntersection of subspaces

If $V$ and $W$ are subvector spaces of $\RNrSpc{n}$, then their intersection $\Intrsctn{V}{W}$ is also a subvector space of $\RNrSpc{n}$.

Subvector Spaces

Introduction

Study Materials