Finding the double straight lines of transformation?
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HallsofIvy
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I'm not sure what you mean by "the double straight lines" but I suspect you need to look for the eigenvectors of this linear transformation. This transformation can be written as the matrixCan you help me find the double straight lines of the transformation φ:
{px`1=4x1+2x2
px`2=6x2
px`3=2x1+4x3
or the basic method of finding it of any transformation? thanks
\(\displaystyle \begin{bmatrix}4 & 2 & 0 \\ 0 & 6 & 0 \\ 2 & 0 & 4\end{bmatrix}\)
Its "characteristic equation" is
\(\displaystyle \left|\begin{matrix}4-\lambda & 2 & 0 \\ 0 & 6-\lambda & 0 \\ 2 & 0 & 4- \lambda\end{matrix}\right|\)
expanding on the second row,
\(\displaystyle (6- \lambda)\left|\begin{matrix}4-\lambda & 0 \\ 2 & 4- \lambda\end{matrix}\right|= (6- \lambda)(4-\lambda)^2= 0\)
which has 4 as a double eigenvalue (and 6 as the other eigenvalue). Perhaps the "double straight lines" refers to lines in the direction of the eigenvectors corresponding to eigenvalue 4?
this problem is for a project i have to make which can level my grade, so in class we haven't mentioned it (unless i missed something)... but it's probably what HallsofIvy explained... i don't know if i had explained correctly everything, because the translation from my language is almost literal and maybe could have made some mistakes.
There are "eigenvectors" of the transformation, such that when the transformation is applied to an eigenvector, the result is to multiply the original vector by a constant (called an "eigenvalue" of the transformation). That is, the resulting vector is parallel to the original direction. Could that be what is meant by a double straight line? If so, HallsofIvy has shown how to find the eigenvalues (4, 6, and 4). Then for instance for \(\displaystyle \lambda = 6\), solve the linear system
\(\displaystyle \begin{bmatrix} -2 & 2 & 0 \\ 0 & 0 & 0 \\ 2 & 0 & -2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = 0 \)
By inspection, a solution is \(\displaystyle \begin{bmatrix} 1 \\1 \\ 1\end{bmatrix}\)
Any vector parallel to this direction remains parallel when the transformation is applied, with magnitude multiplied by 6.
A note about language. When there is any question about translation, I (for one) would like to see the question as written in your own language, as well as your English translation. That gives us a chance to see where literal translation fails, and also teaches us something about how mathematical operations are expressed (which may be hard to find just using a dictionary!).
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thanks for the help 
we've actually learned eigenvectors and called them proper vector in my language... so for your information, (i don't know if you know Bulgarian ) here's it on my language: Спрямо проективна координатна система е дадена линейната трансформация φ. Да се намерят двойните прави на φ.
we've actually learned eigenvectors and called them proper vector in my language... so for your information, (i don't know if you know Bulgarian ) here's it on my language: Спрямо проективна координатна система е дадена линейната трансформация φ. Да се намерят двойните прави на φ.Thanks - that was a "fun" exercise. I forget how many people in the world do NOT use the Roman alphabet. I have a visitor from the Czech Republic who was able to pronounce the words in Cyrillic script, but she didn't recognize many of the words. The most obvious transliterations were "система" and "трансформация". Thanks anyway - maybe some other users or tutors will be able to read it!thanks for the help