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SectionSSSpanning Sets

In this section we will provide an extremely compact way to describe an infinite set of vectors, making use of linear combinations. This will give us a convenient way to describe the solution set of a linear system, the null space of a matrix, and many other sets of vectors.

SubsectionSSVSpan of a Set of Vectors

In Example VFSAL we saw the solution set of a homogeneous system described as all possible linear combinations of two particular vectors. This is a useful way to construct or describe infinite sets of vectors, so we encapsulate the idea in a definition.

DefinitionSSCVSpan of a Set of Column Vectors

Given a set of vectors \(S=\{\vectorlist{u}{p}\}\text{,}\) their span, \(\spn{S}\text{,}\) is the set of all possible linear combinations of \(\vectorlist{u}{p}\text{.}\) Symbolically, \begin{align*} \spn{S}&=\setparts{\lincombo{\alpha}{u}{p}}{\alpha_i\in\complexes,\,1\leq i\leq p}\\ &=\setparts{\sum_{i=1}^{p}\alpha_i\vect{u}_i}{\alpha_i\in\complexes,\,1\leq i\leq p}\text{.} \end{align*}

The span is just a set of vectors, though in all but one situation it is an infinite set. (Just when is it not infinite? See Exercise SS.T30.) So we start with a finite collection of vectors \(S\) (\(p\) of them to be precise), and use this finite set to describe an infinite set of vectors, \(\spn{S}\text{.}\) Confusing the finite set \(S\) with the infinite set \(\spn{S}\) is one of the most persistent problems in understanding introductory linear algebra. We will see this construction repeatedly, so let us work through some examples to get comfortable with it. The most obvious question about a set is if a particular item of the correct type is in the set, or not in the set.

Having analyzed Archetype A in Example SCAA, we will of course subject Archetype B to a similar investigation.

SubsectionSSNSSpanning Sets of Null Spaces

We saw in Example VFSAL that when a system of equations is homogeneous the solution set can be expressed in the form described by Theorem VFSLS where the vector \(\vect{c}\) is the zero vector. We can essentially ignore this vector, so that the remainder of the typical expression for a solution looks like an arbitrary linear combination, where the scalars are the free variables and the vectors are \(\vectorlist{u}{n-r}\text{.}\) Which sounds a lot like a span. This is the substance of the next theorem.

Proof

Notice that the hypotheses of Theorem VFSLS and Theorem SSNS are slightly different. In the former, \(B\) is the row-reduced version of an augmented matrix of a linear system, while in the latter, \(B\) is the row-reduced version of an arbitrary matrix. Understanding this subtlety now will avoid confusion later.

Here is an example that will simultaneously exercise the span construction and Theorem SSNS, while also pointing the way to the next section.

SubsectionReading Questions

1

Let S be the set of three vectors below. \begin{equation*} S=\set{\colvector{1\\2\\-1},\,\colvector{3\\-4\\2},\,\colvector{4\\-2\\1}}\text{.} \end{equation*} Let \(W=\spn{S}\) be the span of S. Is the vector \(\colvector{-1\\8\\-4}\) in \(W\text{?}\) Give an explanation of the reason for your answer.

2

Use \(S\) and \(W\) from the previous question. Is the vector \(\colvector{6\\5\\-1}\) in \(W\text{?}\) Give an explanation of the reason for your answer.

3

For the matrix \(A\) below, find a set \(S\) so that \(\spn{S}=\nsp{A}\text{,}\) where \(\nsp{A}\) is the null space of \(A\text{.}\) (See Theorem SSNS.) \begin{equation*} A= \begin{bmatrix} 1 & 3 & 1 & 9\\ 2 & 1 & -3 & 8\\ 1 & 1 & -1 & 5 \end{bmatrix} \end{equation*}

SubsectionExercises

C22

For each archetype that is a system of equations, consider the corresponding homogeneous system of equations. Write elements of the solution set to these homogeneous systems in vector form, as guaranteed by Theorem VFSLS. Then write the null space of the coefficient matrix of each system as the span of a set of vectors, as described in Theorem SSNS.

Archetype A, Archetype B, Archetype C, Archetype D/Archetype E, Archetype F, Archetype G/Archetype H, Archetype I, Archetype J

Solution
C23

Archetype K and Archetype L are defined as matrices. Use Theorem SSNS directly to find a set \(S\) so that \(\spn{S}\) is the null space of the matrix. Do not make any reference to the associated homogeneous system of equations in your solution.

Solution
C40

Suppose that \(S=\set{\colvector{2\\-1\\3\\4},\,\colvector{3\\2\\-2\\1}}\text{.}\) Let \(W=\spn{S}\) and let \(\vect{x}=\colvector{5\\8\\-12\\-5}\text{.}\) Is \(\vect{x}\in W\text{?}\) If so, provide an explicit linear combination that demonstrates this.

Solution
C41

Suppose that \(S=\set{\colvector{2\\-1\\3\\4},\,\colvector{3\\2\\-2\\1}}\text{.}\) Let \(W=\spn{S}\) and let \(\vect{y}=\colvector{5\\1\\3\\5}\text{.}\) Is \(\vect{y}\in W\text{?}\) If so, provide an explicit linear combination that demonstrates this.

Solution
C42

Suppose \(R=\set{\colvector{2\\-1\\3\\4\\0},\,\colvector{1\\1\\2\\2\\-1},\,\colvector{3\\-1\\0\\3\\-2}}\text{.}\) Is \(\vect{y}=\colvector{1\\-1\\-8\\-4\\-3}\) in \(\spn{R}\text{?}\)

Solution
C43

Suppose \(R=\set{\colvector{2\\-1\\3\\4\\0},\,\colvector{1\\1\\2\\2\\-1},\,\colvector{3\\-1\\0\\3\\-2}}\text{.}\) Is \(\vect{z}=\colvector{1\\1\\5\\3\\1}\) in \(\spn{R}\text{?}\)

Solution
C44

Suppose that \begin{equation*} S=\set{ \colvector{-1 \\ 2 \\ 1},\, \colvector{ 3 \\ 1 \\ 2},\, \colvector{ 1 \\ 5 \\ 4},\, \colvector{-6 \\ 5 \\ 1} }\text{.} \end{equation*} Let \(W=\spn{S}\) and let \(\vect{y}=\colvector{-5\\3\\0}\text{.}\) Is \(\vect{y}\in W\text{?}\) If so, provide an explicit linear combination that demonstrates this.

Solution
C45

Suppose that \begin{equation*} S=\set{ \colvector{-1 \\ 2 \\ 1},\, \colvector{ 3 \\ 1 \\ 2},\, \colvector{ 1 \\ 5 \\ 4},\, \colvector{-6 \\ 5 \\ 1} }\text{.} \end{equation*} Let \(W=\spn{S}\) and let \(\vect{w}=\colvector{2\\1\\3}\text{.}\) Is \(\vect{w}\in W\text{?}\) If so, provide an explicit linear combination that demonstrates this.

Solution
C50

Let \(A\) be the matrix below.

  1. Find a set \(S\) so that \(\nsp{A}=\spn{S}\text{.}\)
  2. If \(\vect{z}=\colvector{3 \\ -5 \\ 1 \\ 2}\text{,}\) then show directly that \(\vect{z}\in\nsp{A}\text{.}\)
  3. Write \(\vect{z}\) as a linear combination of the vectors in \(S\text{.}\)

\begin{align*} A= \begin{bmatrix} 2 & 3 & 1 & 4 \\ 1 & 2 & 1 & 3 \\ -1 & 0 & 1 & 1 \end{bmatrix} \end{align*}

Solution
C60

For the matrix \(A\) below, find a set of vectors \(S\) so that the span of \(S\) equals the null space of \(A\text{,}\) \(\spn{S}=\nsp{A}\text{.}\) \begin{equation*} A= \begin{bmatrix} 1 & 1 & 6 & -8\\ 1 & -2 & 0 & 1\\ -2 & 1 & -6 & 7 \end{bmatrix} \end{equation*}

Solution
M10

Consider the set of all size \(2\) vectors in the Cartesian plane \(\real{2}\text{.}\)

  1. Give a geometric description of the span of a single vector.
  2. How can you tell if two vectors span the entire plane, without doing any row reduction or calculation?
Solution
M11

Consider the set of all size \(3\) vectors in Cartesian 3-space \(\real{3}\text{.}\)

  1. Give a geometric description of the span of a single vector.
  2. Describe the possibilities for the span of two vectors.
  3. Describe the possibilities for the span of three vectors.
Solution
M12

Let \(\vect{u} = \colvector{1\\3\\-2}\) and \(\vect{v} = \colvector{2\\-2\\1}\text{.}\)

  1. Find a vector \(\vect{w}_1\text{,}\) different from \(\vect{u}\) and \(\vect{v}\text{,}\) so that \(\spn{\set{\vect{u}, \vect{v}, \vect{w}_1}} = \spn{\set{\vect{u}, \vect{v}}}\text{.}\)
  2. Find a vector \(\vect{w}_2\) so that \(\spn{\set{\vect{u}, \vect{v}, \vect{w}_2}} \ne \spn{\set{\vect{u}, \vect{v}}}\text{.}\)
Solution
M20

In Example SCAD we began with the four columns of the coefficient matrix of Archetype D, and used these columns in a span construction. Then we methodically argued that we could remove the last column, then the third column, and create the same set by just doing a span construction with the first two columns. We claimed we could not go any further, and had removed as many vectors as possible. Provide a convincing argument for why a third vector cannot be removed.

M21

In the spirit of Example SCAD, begin with the four columns of the coefficient matrix of Archetype C, and use these columns in a span construction to build the set \(S\text{.}\) Argue that \(S\) can be expressed as the span of just three of the columns of the coefficient matrix (saying exactly which three) and in the spirit of Exercise SS.M20 argue that no one of these three vectors can be removed and still have a span construction create \(S\text{.}\)

Solution
T10

Suppose that \(\vect{v}_1,\,\vect{v}_2\in\complex{m}\text{.}\) Prove that \begin{equation*} \spn{\set{\vect{v}_1,\,\vect{v}_2}}=\spn{\set{\vect{v}_1,\,\vect{v}_2,\,5\vect{v}_1+3\vect{v}_2}} \end{equation*}

Solution
T20

Suppose that \(S\) is a set of vectors from \(\complex{m}\text{.}\) Prove that the zero vector, \(\zerovector\text{,}\) is an element of \(\spn{S}\text{.}\)

Solution
T21

Suppose that \(S\) is a set of vectors from \(\complex{m}\) and \(\vect{x},\,\vect{y}\in\spn{S}\text{.}\) Prove that \(\vect{x}+\vect{y}\in\spn{S}\text{.}\)

T22

Suppose that \(S\) is a set of vectors from \(\complex{m}\text{,}\) \(\alpha\in\complexes\text{,}\) and \(\vect{x}\in\spn{S}\text{.}\) Prove that \(\alpha\vect{x}\in\spn{S}\text{.}\)

T30

For which sets \(S\) is \(\spn{S}\) a finite set? Give a proof for your answer.

Solution