How do I find the matrix A to the linear transformation T: R^3 --> R^3
it's defined by;
1. reflection against 3x - 6y + 5z = 0
2. projection onto 2x + 6y + 4z = 0
Here are the formulas that you need (I won't do the actual question for you).
Suppose that is a unit vector (so that ). Then the projection onto the one-dimensional subspace spanned by n is .
If px + qy + rz = 0 is the equation of a plane, let n be a unit vector orthogonal to the plane. So . Then the matrix of the projection onto the plane is , and the matrix of the reflection in the plane is .
To find the matrix for the composition of two such operations, form the matrices for each operation, then multiply them. So the matrix for reflection in 3x - 6y + 5z = 0 followed by projection onto 2x + 6y + 4z = 0 is , where m and n are the normalised versions of (3,-6,5) and (2,6,4) respectively.