Let \(A\) be any \(2 \times 3\) matrix such that \({\mathop{\rm rank}\nolimits} A = 1\), let u be the first column of \(A\), and suppose \({\mathop{\rm u}\nolimits} \ne 0\). Explain why there is a vector v in \({\mathbb{R}^3}\) such that \(A = {{\mathop{\rm uv}\nolimits} ^T}\). How could this construction be modified if the first column of \(A\) were zero?

Suppose \(A = \left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm u}\nolimits} _1}}&{{{\mathop{\rm u}\nolimits} _2}}&{{{\mathop{\rm u}\nolimits} _3}}\end{array}} \right]\) and rank \(A = 1\). Also, assume that \({{\mathop{\rm u}\nolimits} _1} \ne 0\). Then, \(\left\{ {{{\mathop{\rm u}\nolimits} _1}} \right\}\) is the basis for \({\mathop{\rm Col}\nolimits} A\) because it is assumed to be one-dimensional.

Therefore, there are two scalars \({\mathop{\rm x}\nolimits} \) and \(y\) with \({{\mathop{\rm u}\nolimits} _2} = {\mathop{\rm x}\nolimits} {{\mathop{\rm u}\nolimits} _1}\), \({{\mathop{\rm u}\nolimits} _3} = {\mathop{\rm y}\nolimits} {{\mathop{\rm u}\nolimits} _1}\), and \(A = {{\mathop{\rm u}\nolimits} _1}{{\mathop{\rm v}\nolimits} ^T}\), where \({\mathop{\rm v}\nolimits} = \left[ {\begin{array}{*{20}{c}}1\\x\\y\end{array}} \right]\).

When \({{\mathop{\rm u}\nolimits} _1} = 0\), \({{\mathop{\rm u}\nolimits} _2} \ne 0\). Similarly, \(\left\{ {{{\mathop{\rm u}\nolimits} _1}} \right\}\) is a basis for \({\mathop{\rm Col}\nolimits} A\) because \({\mathop{\rm Col}\nolimits} A\) is assumed to be one-dimensional. Therefore, there is a scalar \({\mathop{\rm x}\nolimits} \) with \({{\mathop{\rm u}\nolimits} _3} = {\mathop{\rm x}\nolimits} {{\mathop{\rm u}\nolimits} _2}\) and \(A = {{\mathop{\rm u}\nolimits} _2}{{\mathop{\rm v}\nolimits} ^T}\), where \({\mathop{\rm v}\nolimits} = \left[ {\begin{array}{*{20}{c}}0\\1\\{\mathop{\rm x}\nolimits} \end{array}} \right]\).

When \({{\mathop{\rm u}\nolimits} _1} = {{\mathop{\rm u}\nolimits} _2} = 0\) and \({{\mathop{\rm u}\nolimits} _3} \ne 0\), then \(A = {{\mathop{\rm u}\nolimits} _3}{{\mathop{\rm v}\nolimits} ^T}\), where \({\mathop{\rm v}\nolimits} = \left[ {\begin{array}{*{20}{c}}0\\0\\1\end{array}} \right]\).

Thus, there is a vector in \({\mathbb{R}^3}\) such that \(A = {{\mathop{\rm uv}\nolimits} ^T}\).