• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon

Suggested languages for you:

Americas

Europe

Q31E

Expert-verified
Found in: Page 191

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# 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 $${\mathop{\rm v}\nolimits} = \left[ {\begin{array}{*{20}{c}}a\\b\\c\end{array}} \right]$$.

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

See the step by step solution

## Step 1: 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}

## Step 2: 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 $${{\mathop{\rm uv}\nolimits} ^T} \le 1$$.