I have no idea how to solve it. Please help me out! Thank you!

Let $\alpha$ be a plane in $\mathbb{R}^3$ passing through the origin, suppose $\alpha$ is given by the equation $$ax + by + cz = 0.$$ Reflection in $\alpha$ is a linear transformation $T$ of $\mathbb{R}^3$.

Find a matrix representation of $T$ with respect to the standard basis of $\mathbb{R}^3$ (bear in mind that reflection does not change length).

Hint: find $T$ as a composition of a transformation, which maps the normal of $\alpha$ to one of the coordinate axes, and a reflection in the corresponding coordinate plane.

  • $\begingroup$ Consider an ordered basis of $\mathbb{R}^3$ of the form $(a,b,c)$, $(-b,a,0)$ and $(0,-c,b)$ (the first vector normal to the plane and the last two in the plane). With respect a reflection matrix will have the form $$M = \left[ {\begin{array}{ccc} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} } \right]$$ and after this, write the vectors with canonical coordintes in the given basis of $\mathbb{R}^3$, $(a,b,c)$, $(-b,a,0)$ and $(0,-c,b)$. In this way I think you can get something (perhaps if $b=0$ you might change something on the choice of vector on the plane). $\endgroup$ – matgaio Apr 22 '12 at 2:08
  • $\begingroup$ Thanks for answering! Isn't the basis should be the standard basis (1 0 0)(0 1 0)(0 0 1)? $\endgroup$ – Megan Apr 22 '12 at 3:30
  • $\begingroup$ The point, Megan, is that if you have the matrix with respect to one basis (like matgaio's), there is a standard procedure for finding the matrix with respect to another basis (like the standard one). $\endgroup$ – Gerry Myerson Apr 22 '12 at 4:43
  • $\begingroup$ You are welcome, @Megan. I've taken that basis because of the geometrical meaning of the transformation. Think on this: a reflection with respect to a plane must send the normal $N$ of the plane in $-N$ and must left the plane itself invariant. So, if we take any basis of the plane, say $U$ and $V$, we will have $T(N)=-N$, $T(U)=U$ and $T(V)=V$, and then the matrix with respect to this basis is that I've writen above. The question you will have to work on once you get this facts is how to change the given basis to the canonical basis. $\endgroup$ – matgaio Apr 22 '12 at 5:00

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