Respuesta :
The basis of [tex]{W^ \bot }[/tex] is [tex]\boxed{{\text{span}}\left\{ {\left[ {\begin{array}{*{20}{c}}{ - 1}\\{ - 1}\\1 \end{array}} \right]} \right\}}[/tex]
Further explanation:
Given:
The vector is,
[tex]\left[ {\begin{array}{*{20}{c}}x\\y\\{x + y} \end{array}} \right][/tex]
Explanation:
Consider the set of all vectors can be expressed as follows,
[tex]\left[{\begin{array}{*{20}{c}}x\\y\\ {x + y} \end{array}} \right] = x\left[ {\begin{array}{*{20}{c}}1\\0\\1\end{array}} \right] + y\left[ {\begin{array}{*{20}{c}}0\\1\\1\end{array}} \right][/tex]
The spanned vectors of [tex]W[/tex] are [tex]\left[ {\begin{array}{*{20}{c}}1 \\ 0\\1 \end{array}} \right]{\text{ and }}\left[ {\begin{array}{*{20}{c}}0 \\1 \\1 \end{array}} \right][/tex]
Consider a vector [tex]\left[ {\begin{array}{*{20}{c}}p\\q\\r\end{array}} \right][/tex] as [tex]{W^ \bot }.[/tex]
The dot product of [tex]W[/tex] and [tex]{W^ \bot }[/tex] must be zero.
[tex]\begin{aligned}\left[{\begin{array}{*{20}{c}}1\\0\\1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}}p\\q \\r\end{array}} \right]&= 0 \hfill \\\left[ {\begin{array}{*{20}{c}}1&0&1 \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}}p \\q\\r\end{array}} \right] &= 0 \hfill\\\end{aligned}[/tex]
Further solve the above equation,
[tex]\begin{aligned}p + r &= 0\\p&= - r\\\end{aligned}[/tex]
[tex]\begin{aligned}\left[{\begin{array}{*{20}{c}}0\\1\\1\end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}}p\\q\\r\end{array}} \right]&= 0\\\left[{\begin{array}{*{20}{c}}0&1&1\end{array}} \right] \cdot \left[{\begin{array}{*{20}{c}}p\\q\\r\end{array}} \right]&= 0\\\end{aligned}[/tex]
Further solve the above equation.
[tex]\begin{aligned}q + r &=0\\q &= - r\\\end{aligned}[/tex]
Therefore, [tex]p = q.[/tex]
The matrix [tex]{W^ \bot }[/tex] will be [tex]\left[ {\begin{array}{*{20}{c}}{ - c}\\ { - c}\\c\end{array}} \right][/tex]
The basis of can be obtained as follows,
[tex]\left[{\begin{array}{*{20}{c}}{ - c}\\{ - c}\\c\end{array}} \right] = c\left[ {\begin{array}{*{20}{c}}{ - 1}\\{ - 1}\\1\end{array}} \right][/tex]
The basis of [tex]{W^ \bot }[/tex] is [tex]\boxed{{\text{span}}\left\{ {\left[ {\begin{array}{*{20}{c}}{ - 1} \\{ - 1}\\1 \end{array}} \right]} \right\}}[/tex]
Learn more:
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3. Learn more about range and domain of the function https://brainly.com/question/3412497
Answer details:
Grade: College
Subject: Mathematics
Chapter: Vectors and matrices
Keywords: W set, all vectors, x, y, x + y, real numbers, perpendicular, matrices, vectors, basis.
