The basis of
is ![\boxed{{\text{span}}\left\{ {\left[ {\begin{array}{*{20}{c}}{ - 1}\\{ - 1}\\1 \end{array}} \right]} \right\}}](https://tex.z-dn.net/?f=%5Cboxed%7B%7B%5Ctext%7Bspan%7D%7D%5Cleft%5C%7B%20%7B%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%7B%20-%201%7D%5C%5C%7B%20-%201%7D%5C%5C1%20%5Cend%7Barray%7D%7D%20%5Cright%5D%7D%20%5Cright%5C%7D%7D)
Further explanation:
Given:
The vector is,
![\left[ {\begin{array}{*{20}{c}}x\\y\\{x + y} \end{array}} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7Dx%5C%5Cy%5C%5C%7Bx%20%2B%20y%7D%20%5Cend%7Barray%7D%7D%20%5Cright%5D)
Explanation:
Consider the set of all vectors can be expressed as follows,
![\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]](https://tex.z-dn.net/?f=%5Cleft%5B%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7Dx%5C%5Cy%5C%5C%20%7Bx%20%2B%20y%7D%20%5Cend%7Barray%7D%7D%20%5Cright%5D%20%3D%20x%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D1%5C%5C0%5C%5C1%5Cend%7Barray%7D%7D%20%5Cright%5D%20%2B%20y%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D0%5C%5C1%5C%5C1%5Cend%7Barray%7D%7D%20%5Cright%5D)
The spanned vectors of
are ![\left[ {\begin{array}{*{20}{c}}1 \\ 0\\1 \end{array}} \right]{\text{ and }}\left[ {\begin{array}{*{20}{c}}0 \\1 \\1 \end{array}} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D1%20%5C%5C%200%5C%5C1%20%5Cend%7Barray%7D%7D%20%5Cright%5D%7B%5Ctext%7B%20and%20%7D%7D%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D0%20%5C%5C1%20%5C%5C1%20%5Cend%7Barray%7D%7D%20%5Cright%5D)
Consider a vector
as 
The dot product of
and
must be zero.
![\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}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cleft%5B%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D1%5C%5C0%5C%5C1%20%5Cend%7Barray%7D%7D%20%5Cright%5D%20%5Ccdot%20%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7Dp%5C%5Cq%20%5C%5Cr%5Cend%7Barray%7D%7D%20%5Cright%5D%26%3D%200%20%5Chfill%20%5C%5C%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D1%260%261%20%5Cend%7Barray%7D%7D%20%5Cright%5D%20%5Ccdot%20%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7Dp%20%5C%5Cq%5C%5Cr%5Cend%7Barray%7D%7D%20%5Cright%5D%20%26%3D%200%20%5Chfill%5C%5C%5Cend%7Baligned%7D)
Further solve the above equation,

![\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}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cleft%5B%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D0%5C%5C1%5C%5C1%5Cend%7Barray%7D%7D%20%5Cright%5D%20%5Ccdot%20%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7Dp%5C%5Cq%5C%5Cr%5Cend%7Barray%7D%7D%20%5Cright%5D%26%3D%200%5C%5C%5Cleft%5B%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D0%261%261%5Cend%7Barray%7D%7D%20%5Cright%5D%20%5Ccdot%20%5Cleft%5B%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7Dp%5C%5Cq%5C%5Cr%5Cend%7Barray%7D%7D%20%5Cright%5D%26%3D%200%5C%5C%5Cend%7Baligned%7D)
Further solve the above equation.

Therefore, 
The matrix
will be ![\left[ {\begin{array}{*{20}{c}}{ - c}\\ { - c}\\c\end{array}} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%7B%20-%20c%7D%5C%5C%20%7B%20-%20c%7D%5C%5Cc%5Cend%7Barray%7D%7D%20%5Cright%5D)
The basis of can be obtained as follows,
![\left[{\begin{array}{*{20}{c}}{ - c}\\{ - c}\\c\end{array}} \right] = c\left[ {\begin{array}{*{20}{c}}{ - 1}\\{ - 1}\\1\end{array}} \right]](https://tex.z-dn.net/?f=%5Cleft%5B%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%7B%20-%20c%7D%5C%5C%7B%20-%20c%7D%5C%5Cc%5Cend%7Barray%7D%7D%20%5Cright%5D%20%3D%20c%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%7B%20-%201%7D%5C%5C%7B%20-%201%7D%5C%5C1%5Cend%7Barray%7D%7D%20%5Cright%5D)
The basis of
is ![\boxed{{\text{span}}\left\{ {\left[ {\begin{array}{*{20}{c}}{ - 1} \\{ - 1}\\1 \end{array}} \right]} \right\}}](https://tex.z-dn.net/?f=%5Cboxed%7B%7B%5Ctext%7Bspan%7D%7D%5Cleft%5C%7B%20%7B%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7B%2A%7B20%7D%7Bc%7D%7D%7B%20-%201%7D%20%5C%5C%7B%20-%201%7D%5C%5C1%20%5Cend%7Barray%7D%7D%20%5Cright%5D%7D%20%5Cright%5C%7D%7D)
Learn more:
1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function 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.