2.3 Span of a Set of Vectors
Reading
Try out the preview activity and read Chapter 2 Section 3 in Understanding Linear Algebra by David Austin.
Definition: The span of a set of vectors [latex]\{ \vec{v}_1,\vec{v}_2, \dots, \vec{v}_k\ \}[/latex] is the set of all linear combinations [latex]\{ c_1\vec{v}_1+c_2\vec{v}_2+ \dots+c_k\vec{v}_k\ | c_1, \dots, c_k \in \mathbb{R} \}[/latex]. In other words, the span is all of the vectors that you can build by scaling and summing the vectors [latex]\{ \vec{v}_1,\vec{v}_2, \dots, \vec{v}_k\ \}[/latex]
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We begin with a few examples of checking if a vector is in the span of a set of vectors.
Finally we look at ways to describe the span of a set of vectors:
Proposition
The set [latex]\{ \vec{v}_1,\vec{v}_2, \dots, \vec{v}_k\ \}[/latex] of vectors in [latex]\mathbb{R}^n[/latex] will span [latex]\mathbb{R}^n[/latex] only if the reduced row echelon form of the matrix [latex]A= [ \vec{v}_1\vec{v}_2 \dots \vec{v}_k ][/latex] has a pivot in each row.
Consider:
Is the vector [latex]\vec{v}=\begin{bmatrix} \pi\\e\end{bmatrix}[/latex] in the span of {[latex]\begin{bmatrix} 1\\ 2 \end{bmatrix}[/latex], [latex]\begin{bmatrix} 2 \\1 \end{bmatrix}[/latex]}? Describe this span.
Is the vector [latex]\vec{v}= \begin{bmatrix} 2 \\ 3 \\ 5 \end{bmatrix}[/latex] in the span of {[latex]\begin{bmatrix} 4 \\ 6 \\ 10 \end{bmatrix}[/latex], [latex]\begin{bmatrix} 1 \\ 0 \\ 2\end{bmatrix}[/latex]}? Describe this span.
