Skip to main content\(\newcommand{\avec}{{\mathbf a}}
\newcommand{\bvec}{{\mathbf b}}
\newcommand{\cvec}{{\mathbf c}}
\newcommand{\dvec}{{\mathbf d}}
\newcommand{\dtil}{\widetilde{\mathbf d}}
\newcommand{\evec}{{\mathbf e}}
\newcommand{\fvec}{{\mathbf f}}
\newcommand{\nvec}{{\mathbf n}}
\newcommand{\pvec}{{\mathbf p}}
\newcommand{\qvec}{{\mathbf q}}
\newcommand{\svec}{{\mathbf s}}
\newcommand{\tvec}{{\mathbf t}}
\newcommand{\uvec}{{\mathbf u}}
\newcommand{\vvec}{{\mathbf v}}
\newcommand{\wvec}{{\mathbf w}}
\newcommand{\xvec}{{\mathbf x}}
\newcommand{\yvec}{{\mathbf y}}
\newcommand{\zvec}{{\mathbf z}}
\newcommand{\rvec}{{\mathbf r}}
\newcommand{\mvec}{{\mathbf m}}
\newcommand{\zerovec}{{\mathbf 0}}
\newcommand{\onevec}{{\mathbf 1}}
\newcommand{\real}{{\mathbb R}}
\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2
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#4 \\ #5 \\ \end{array}\right]}
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#4 \\ #5 \\ \end{array}\right]}
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\newcommand{\laspan}[1]{\text{Span}\{#1\}}
\newcommand{\bcal}{{\cal B}}
\newcommand{\ccal}{{\cal C}}
\newcommand{\scal}{{\cal S}}
\newcommand{\wcal}{{\cal W}}
\newcommand{\ecal}{{\cal E}}
\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}
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\newcommand{\lgray}[1]{\color{lightgray}{#1}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\row}{\text{Row}}
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\renewcommand{\row}{\text{Row}}
\newcommand{\nul}{\text{Nul}}
\newcommand{\var}{\text{Var}}
\newcommand{\corr}{\text{corr}}
\newcommand{\len}[1]{\left|#1\right|}
\newcommand{\bbar}{\overline{\bvec}}
\newcommand{\bhat}{\widehat{\bvec}}
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\newcommand{\xhat}{\widehat{\xvec}}
\newcommand{\vhat}{\widehat{\vvec}}
\newcommand{\uhat}{\widehat{\uvec}}
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\)
Chapter 6 Span
We now turn to our two fundamental questions rephrased here in terms of matrix multiplication.
Existence: Is there a solution to the equation \(A\xvec=\bvec\text{?}\)
Uniqueness: If there is a solution to the equation \(A\xvec=\bvec\text{,}\) is it unique?
In this section, we focus on the existence question and see how it leads to the concept of the span of a set of vectors.
