Spans and Subspaces

Section 3.1 Spans and Subspaces

Subsection 3.1.1 Definitions

This chapter also introduces linear combinations, spans and linear dependence. These are some of the most important and central definitions in linear algebra. Let me start by defining the these three important terms.

Definition 3.1.1.

A linear combination of a set of vectors \(\{v_1, v_2, \ldots, v_k\}\) is a sum of the form \(a_1 v_1 + a_2 v_2 + \ldots a_k v_k\) where the \(a_i \in \RR\text{.}\)

Definition 3.1.2.

The span of a set of vectors \(\{v_1, v_2, \ldots, v_k\}\text{,}\) written

\begin{equation*} \Span \{v_1, v_2, \ldots, v_k\}\text{,} \end{equation*}

is the set of all possible linear combinations of the vectors.

Spans have an important geometric interpretation, but I’ll get to that in the next section.

Definition 3.1.3.

A set of vectors \(\{v_1, v_2, \ldots, v_k\}\) in \(\RR^n\) is called linearly independent if the equation

\begin{equation*} a_1 v_1 + a_2 v_2 + a_3 v_3 + \ldots + a_k v_k = 0 \end{equation*}

has only the trivial solution: for all \(i\text{,}\) \(a_i = 0\text{.}\) If a set of vectors isn’t linearly independent, it is called linearly dependent.

This may seem like a strange definition, but it algebraically captures the idea of independent directions. A set of vectors is linearly independent if all of them point in fundamentally different directions. We could also say that a set of vectors is linearly independent if no vector is in the span of the other vectors. No vector is a redundant piece of information; if we remove any vectors, the span of the set gets smaller.

In order for a set like this to be linearly independent, I need \(k \leq n\text{.}\) \(\RR^n\) has only \(n\) independent directions, so it is impossible to have more than \(n\) linearly independent vectors in \(\RR^n\text{.}\)

Subsection 3.1.2 Linear and Affine Subspaces

Using the language of linear combinations and spans, I can define the major objects that live in Euclidean space.

Definition 3.1.4.

A linear subspace of \(\RR^n\) is a non-empty set of vectors \(L\) which satisfies the following two properties.

If \(u,v \in L\) then \(u+v \in L\text{.}\) The term for this property is: closed under vector addition.
If \(u \in L\) and \(a \in \RR\) then \(au \in L\text{.}\) The term for this property is: closed under scalar multiplication.

There are two basic operations on \(\RR^n\text{:}\) adding vectors and multipling vectdors by scalars. Linear subspaces are just subsets where both operations stay within the subset.

Geometrically, vector addition and scalar multiplication produce flat objects: lines, planes, and their higher-dimensional analogues. Also, since \(a=0\) is possible, it must be true that \(0 \in L\text{.}\) So linear subspaces can be informally defined as ‘flat’ subsets which include the origin. If I relax the definition slighlty to consider ‘flat’ subsets which do not contain the origin, I get a related definition.

Definition 3.1.5.

An affine subspace of \(\RR^n\) is a non-empty set of vectors \(A\) which can be described as a sum \(v+u\) where \(v\) is a fixed vector and \(u\) is any vector in some fixed linear subspace \(L\text{.}\) With some abuse of notation, this sum of a fixed vector and a linear subspace is often written

\begin{equation*} A = v + L\text{.} \end{equation*}

Affine subspaces are flat spaces that may be offset from the origin. The vector \(v\) is called the offset vector. Affine spaces include linear spaces since \(v = 0 \) is always possible, leading to \(A=L\text{.}\) Affine objects are the lines, planes and higher-dimensional flat objects that may or may not pass through the origin.

Notice that I defined both affine and linear subspaces to be non-empty. The empty set \(\emptyset\) is not a linear or affine subspace. The smallest linear subspace is \(\{0\}\text{:}\) just the origin. The smallest affine subspace is any isolated point.

Now let me connect the two idea presented so far in this section.

Proposition 3.1.6.

Any span is a linear subspace. Conversely, any linear subspace can be written as a span.

By this proposition, spans provide an algebraic description of the geometric notion of a linear subspace. Since ‘any linear combination’ can include the trivial linear combination where all the real constants are zero, spans always go through the origin. To use spans to define affine subspaces, I have to add an offset vector.

Definition 3.1.7.

An offset span is an affine subspace formed by adding a fixed vector \(u\text{,}\) called the offset vector, to the span of some set of vectors.

Let me talk about spans of small numbers of vectors. First, the span of one non-zero vector is the line (through the origin) consisting of all multiples of that vector. Similarly, I expect the span of two vectors to be a plane. However, here I have a problem: there may be redundant information. For example, in \(\RR^2\text{,}\) we could consider this span.

\begin{equation*} \Span \left\{ \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 4 \end{pmatrix} \right\} \end{equation*}

I would hope the span of two vectors would be the entire plane, but this is just the line in the direction \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\text{.}\) The vector \(\begin{pmatrix} 2 \\ 4 \end{pmatrix}\text{,}\) since it is already a multiple of \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\text{,}\) is redundant.

The problem is magnified in higher dimensions. If I have the span of a large number of vectors in \(\RR^n\text{,}\) it is nearly impossible to tell, at a glance, whether any of the vectors are redundant. I would like to have tools to determine this redundancy. I’m going to define the keys ideas behind this redundancy in the next part of this section, but the calculation tools for checking redundancy have to wait until Section 5.1.

Subsection 3.1.3 Dimension and Basis

I’ve already casually assumed some intuition about dimensions for linear subspaces. A line through the origin should have dimensions one. A plane through the origin should have dimensions two. However, I can’t rely on intuition for mathematical definition: I have to formalize. This second formally defines dimension and related ideas.

Definition 3.1.8.

Let \(L\) be a linear subspace of \(\RR^n\text{.}\) Then \(L\) has dimension k if \(L\) can be written as the span of \(k\) linearly independent vectors.

Definition 3.1.9.

Let \(A\) be an affine subspace of \(\RR^n\) and write \(A\) as \(A = u + L\) for \(L\text{,}\) a linear subspace, and \(u\) an offset vector. Then \(A\) has dimension k if \(L\) has dimension \(k\text{.}\)

This is the proper, complete definition of dimension for linear and affine spaces. It solves the problem of redundant information (either redundant equations for loci or redundant vectors for spans) by insisting on a linearly independent spanning set. It insists that the vectors defining a span are a minimal set. There is a name for such a minimal set.

Definition 3.1.10.

Let \(L\) be a linear subspace of \(\RR^n\text{.}\) A basis for \(L\) is a minimal spanning set; that is, a set of linearly independent vectors that span \(L\text{.}\)

Since a span is the set of all linear combinations, I can think of a basis as a way of writing the vectors of \(L\text{:}\) every vector in \(L\) can be written as a unique linear combination of the basis vectors. A basis gives a nice way to account for all the vectors in \(L\text{.}\)

Linear subspaces have many (infinitely many) different bases. There are some standard choices.

Definition 3.1.11.

The standard basis of \(\RR^2\) is composed of the two-unit vectors in the positive \(x\) and \(y\) directions. I can write any vector as a linear combination of the basis vectors.

\begin{align*} e_1 \amp = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \amp e_2 \amp = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \amp \begin{pmatrix} x \\ y \end{pmatrix} \amp = x e_1 + y e_2 \end{align*}

The standard basis of \(\RR^3\) is composed of the three-unit vectors in the positive \(x\text{,}\) \(y\) and \(z\) directions. I can again write any vector as a linear combination of the basis vectors.

\begin{align*} e_1 \amp = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \amp e_2 \amp = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \amp e_3 \amp = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \amp \begin{pmatrix} x \\ y \\ z \end{pmatrix} \amp = x e_1 + y e_2 + z e_3 \end{align*}

The standard basis of \(\RR^n\) is composed of vectors \(e_1, e_2, \ldots, e_n\) where \(e_i\) has a \(1\) in the \(i\)th component and zeroes in all other components. \(e_i\) is the unit vector in the positive \(i\)th axis direction.

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