Question

Verify that rank \({{\mathop{\rm uv}\nolimits} ^T} \le 1\) if \({\mathop{\rm u}\nolimits} = \left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\5\end{array}} \right]\) and \({\mathop{\rm v}\nolimits} = \left[ {\begin{array}{*{20}{c}}a\\b\\c\end{array}} \right]\).

Short answer

It is verified that rank \({{\mathop{\rm uv}\nolimits} ^T} \le 1\).

Step-by-step solution

Compute the matrix \({{\mathop{\rm uv}\nolimits} ^T}\)

Compute the matrix \({{\mathop{\rm uv}\nolimits} ^T}\) as shown below:

\(\begin{aligned} {{\mathop{\rm uv}\nolimits} ^T} &= \left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\5\end{array}} \right]\left[ {\begin{array}{*{20}{c}}a&b&c\end{array}} \right]\\ &= \left[ {\begin{array}{*{20}{c}}{2a}&{2b}&{2c}\\{ - 3a}&{ - 3b}&{ - 3c}\\{5a}&{5b}&{5c}\end{array}} \right]\end{aligned}\)

Verify that rank \({{\mathop{\rm uv}\nolimits} ^T} \le 1\)

Each column of the matrix \({{\mathop{\rm uv}\nolimits} ^T}\)is a multiple of \({\mathop{\rm u}\nolimits} \).

\(\dim {\mathop{\rm Col}\nolimits} {{\mathop{\rm uv}\nolimits} ^T} = 1\)except when \(a = b = c = 0\) in which \({{\mathop{\rm uv}\nolimits} ^T}\) is a \(3 \times 3\) zero matrix. This means, \(\dim {\mathop{\rm Col}\nolimits} {{\mathop{\rm uv}\nolimits} ^T} = 0\).

However, rank \({{\mathop{\rm uv}\nolimits} ^T} = \dim {\mathop{\rm Col}\nolimits} {{\mathop{\rm uv}\nolimits} ^T} \le 1\).

Thus, it is verified that rank \({{\mathop{\rm uv}\nolimits} ^T} \le 1\).