Mathematics, 19.07.2019 18:40 cdeanrn8971
Let w be the set of all vectors ⎡⎣⎢xyx+y⎤⎦⎥ with x and y real. find a basis of w⊥.
Answers: 1
is ![\boxed{{\text{span}}\left\{ {\left[ {\begin{array}{*{20}{c}}{ - 1}\\{ - 1}\\1 \end{array}} \right]} \right\}}](/tpl/images/0108/8346/ab2b0.png)
![\left[ {\begin{array}{*{20}{c}}x\\y\\{x + y} \end{array}} \right]](/tpl/images/0108/8346/c61e4.png)
![\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]](/tpl/images/0108/8346/64ef1.png)
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]](/tpl/images/0108/8346/f8b12.png)
![\left[ {\begin{array}{*{20}{c}}p\\q\\r\end{array}} \right]](/tpl/images/0108/8346/bfc35.png)
as 
![\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}](/tpl/images/0108/8346/7128f.png)
![\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}](/tpl/images/0108/8346/caf13.png)
![\left[ {\begin{array}{*{20}{c}}{ - c}\\ { - c}\\c\end{array}} \right]](/tpl/images/0108/8346/14f43.png)
![\left[{\begin{array}{*{20}{c}}{ - c}\\{ - c}\\c\end{array}} \right] = c\left[ {\begin{array}{*{20}{c}}{ - 1}\\{ - 1}\\1\end{array}} \right]](/tpl/images/0108/8346/bf794.png)
![\boxed{{\text{span}}\left\{ {\left[ {\begin{array}{*{20}{c}}{ - 1} \\{ - 1}\\1 \end{array}} \right]} \right\}}](/tpl/images/0108/8346/cd7a4.png)
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Let w be the set of all vectors ⎡⎣⎢xyx+y⎤⎦⎥ with x and y real. find a basis of w⊥....
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