Transcript
![Page 1: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/1.jpg)
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![Page 4: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/4.jpg)
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![Page 7: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/7.jpg)
![Page 8: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/8.jpg)
![Page 9: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/9.jpg)
Solution
![Page 10: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/10.jpg)
Solution
![Page 11: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/11.jpg)
![Page 12: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/12.jpg)
![Page 13: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/13.jpg)
Solution
Let us first find an orthogonal basis for W
![Page 14: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/14.jpg)
Now, to avoid fractions in our computation, let us consider 3w2, instead of
w2. Note that the scaling does not change the orthogonality.
We have
Now it suffices to normalize the vectors w1, 3w2 to obtain an orthonormal
basis. Therefore, the set
Problem
![Page 15: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/15.jpg)
Solution
![Page 16: site.iugaza.edu.pssite.iugaza.edu.ps/bhamed/files/2018/09/Examples.pdf · Therefore the formula is given by atl + 2X2 + 3X3 T(x) 4X1 -+ 5X2 -+ 6X3 Since T is a linear transformation,](https://reader034.vdocuments.site/reader034/viewer/2022042019/5e76ed5cfd144453d43bde87/html5/thumbnails/16.jpg)
Given 2x2 +12xy +20y2 is positive definite ?
Solution
The matrix
Is positive definite – its determinant is 4 and its trace is 22 so its eigenvalues are positive
Problem