Let L: R^3 -> R^3 be the linear transformation that is defined by the reflection about the plane P: 2x + y -2z = 0 in R^3. Namely, L(u) = u if u is the vector that lies in the plane P; and L(u) = -u if u is a vector perpendicular to the plane P. Find an orthonormal basis for R^3 and a matrix A such that A is diagonal and A is the matrix representation of L with respect to the orthonormal basis.
The Attempt at a Solution
Basically I tried this problem the only way I knew how, and it was very long and tedious and didn't come out right. I am hoping someone can either (or both) see where I went wrong, or suggest a more streamlined approach.
First off, I was a little unsure by what was meant towards the end of the question about an orthonormal basis. Specifically, is it referring to an orthonormal basis in terms of the domain of L or the range of L?
I chose to assume if was referring to the domain since this fit in with the only way I knew how to approach this problem. The standard basis for R^3 is of course orthonormal. My attempt was to use the theorem that a matrix can be found by applying the transformation to each basis vector, then the columns are the coefficients of this transformation written as a linear combination of the basis vectors of the space being transformed to.
So first off I chose a basis of R^3 in terms of the plane. I choose (1 -2 0) since it satisfies the equation of the plane, the normal vector (2 1 -2), and the cross product of the normal vector and the first vector which is (-4 2 -5). Let's call this set of vectors B
So then I need to reflect the standard basis (let's call it A) across the plane. Rather than calculate out the reflections of A, I thought I could go straight to writing the standard basis as a linear combination of B and then switch the sign of the coefficient of (2 1 -2) since this is the portion (or the projection of the vector onto the normal vector) that will be reflected direction across. In other words, any vector will be equal to its projection into the plane + it's projection onto the normal of the plane. Then reflecting it is changing the sign of the projection onto the normal vector.
So if the matrix C is the B as column vectors:
1 -4 2
-2 2 1
0 -5 -2
I solved Cx = e1, Cx = e2, Cx = e3. (I checked these x's a few times so I am pretty confident about this part). Then I put these x's as the column vectors of a matrix and switched the sign of the last row (again thinking this was writing the reflection of the standard basis across the plane in terms of B). This yielded
18 -36 0
-8 -4 -10
-20 -10 20
all times (1/90). I thought this was the answer but I check multiplying the normal vector but did not get the negative of the normal vector as output.
So, where did I go wrong? And or, what is a streamlined approach. This method seemed way to long, not to mention just incorrect.
PS. I can't figure out how to add matrices officially on this. Is it listed in the Latex Reference button popup?