What is the Matrix of Reflection in Euclidean Space?

[gop]

Homework Statement

V is a three-dimensional euclidean space and v1,v2,v3 is a orthonormal base of that space.
Calculate the Matrix of the reflection over the subspace spanned by v1+v2 and v1+2*v2+3*v3 .

Homework Equations

The Attempt at a Solution

To determine the matrix I have first to select a base I could try to use v1,v2,v3 but I can't see how to determine the entries of the matrix then.
I could use v1+v2 and v1+2*v2+3*v3 (the base of the subspace) and try to extend to a base of R^3; however I can't see how to do that with the general case without knowing what v1,v2,v3 actually is.

 

[Dick]

Why not just write the matrix in the v1,v2,v3 basis? I.e. just treat them as though they were i,j,k. Create an orthonormal basis for the subspace. The basis vectors for it are fixed by the reflection and the orthogonal vector is multiplied by (-1). Once you have it in that basis, then if you really have to, apply the basis change from the standard basis to v1,v2,v3.