Q13E Let \({\mathop{\rm u}\nolimits} ... [FREE SOLUTION]

91影视

Linear Algebra and its Applications

David C. Lay, Steven R. Lay and Judi J. McDonald

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 483 pages

ISBN: 978-03219822384

Chapter 1: Q13E (page 1)

Let ${\mathop{\rm u}\nolimits} = \left( {\begin{array}{{20}{c}}0\\4\\4\end{array}} \right)$ and $A = \left( {\begin{array}{{20}{c}}3&{ - 5}\\{ - 2}&6\\1&1\end{array}} \right)$. Is ${\mathop{\rm u}\nolimits} $ in the plane ${\mathbb{R}^3}$ spanned by the columns of $A$?(see the figure.) Why or why not?

Short Answer

Expert verified

$Ax = u$ has the solution and $u$ is in the plane ${\mathbb{R}^3}$ spanned by the columns of $A$.

Step by step solution

Writing the matrix in the augmented form

The augmented form $\left[ {\begin{array}{*{20}{c}}A&u\end{array}} \right]$ of the given matrix is

$\left[ {\begin{array}{*{20}{c}}3&{ - 5}&0\\{ - 2}&6&4\\1&1&4\end{array}} \right]$.

Step 2:Applying the row operation

Perform an elementary row operationto produce the first augmented matrix.

Interchange rows one and three.

$\left[ {\begin{array}{*{20}{c}}1&1&4\\{ - 2}&6&4\\3&{ - 5}&0\end{array}} \right]$

Step 3:Applying the row operation

Perform an elementary row operationto produce the second augmented matrix.

Write the sum of 2 times row one and row twoin row two.

$\left[ {\begin{array}{*{20}{c}}1&1&4\\0&8&{12}\\3&{ - 5}&0\end{array}} \right]$

Step 4:Applying the row operation

Perform an elementary row operationto produce the third augmented matrix.

Write the sum of $ - 3$ times row one and row threein row three.

$\left[ {\begin{array}{*{20}{c}}1&1&4\\0&8&{12}\\0&{ - 8}&{ - 12}\end{array}} \right]$

Step 5:Applying the row operation

Perform an elementary row operationto produce the fourth augmented matrix.

Write the sum of rowstwo and row threein row three.

$\left[ {\begin{array}{*{20}{c}}1&1&4\\0&8&{12}\\0&0&0\end{array}} \right]$

Converting the matrix in the equation form

To obtain the solution of the vector equation, convert the augmented matrix into vector equations.

Write the obtained matrix $\left[ {\begin{array}{*{20}{c}}1&1&4\\0&8&{12}\\0&0&0\end{array}} \right]$into the equation notation.

$\begin{array}{c}{x_1} + {x_2} = 4\\8{x_2} = 12\end{array}$

Identifying whether $u$ is in the plane${\mathbb{R}^3}$ spanned by the columns of $A$

Every $b$ in ${\mathbb{R}^m}$ is a linear combination of the columns of $A$. A set of vectors $\left\{ {{v_1},...,{v_p}} \right\}$in ${\mathbb{R}^m}$ spans ${\mathbb{R}^m}$ if every vector in ${\mathbb{R}^m}$ is a linear combination of ${v_1},...,{v_p}$. The vector $u$ lies in the plane spanned by the columns of $A$ if and only if it is a linear combination of the columns $A$. This, in turn, occurs if and only if the equation $Ax = u$ has a solution.

Therefore, $Ax = u$ has a solution and $u$ is in the plane ${\mathbb{R}^3}$ spanned by the columns of $A$.