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Section5.2The Span of a Set of Vectors

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SubsectionSpan

It is often convenient to be able to generate the vectors in a vector space via linear combinations from a finite subset of vectors from a vector space.

Definition5.2.1

The span of a set of vectors is the set of all linear combinations of the vectors in that set. If \(S = \{ \vec{v}_1, \ldots, \vec{v}_n \}\) is a set of \(n\) vectors from a vector space, then denote this set \begin{equation*} \linearspan(S) = \{ c_1\vec{v}_1 + \cdots + c_n\vec{v}_n : c_i \in \R, \text{ for } 1 \le i \le n \}. \end{equation*}

For example the span of the set \begin{equation*} \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} \, , \, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\} \end{equation*} is the vector space \(\R^2\) because we can generate any vector in \(\R^2\) like so: \begin{equation*} \begin{bmatrix} a \\ b \end{bmatrix} = a\begin{bmatrix} 1 \\ 0 \end{bmatrix} + b\begin{bmatrix} 0 \\ 1 \end{bmatrix}. \end{equation*}

The term span is used both as a noun and a verb. So we speak of both the span of a set of vectors and how a set spans a vector subspace.

Once we begin to think about subsets of a vector space generating or spanning the whole vector space, a natural question to ask is how many vectors are necessary to generate all vectors in the vector space? Or better yet, what is the smallest set which will still generate or span the vector space?

SubsectionExercises

1

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2

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