EE 203001 Linear Algebra Solutions for Homework # 8 Spring Semester , 2003

12. Let ax + by = 0, a, b ∈ R and a + b 6= 0 be a fixed line in R through the origin. We know u = ( −b √ a2+b2 , a √ a2+b2 ) is a point on line ax + by = 0 and v = ( a √ a2+b2 , b √ a2+b2 ) is a unit vector orthogonal to ( −b √ a2+b2 , a √ a2+b2 ). Thus the set B = {( −b √ a2+b2 , a √ a2+b2 ), ( a √ a2+b2 , b √ a2+b2 )} is an orthonormal basis for R. To find the reflection T (x, y) of (x, y) with respect to the line ax+ by = 0, we consider (x, y) = (−bx+ay √ a2+b2 )u+ ( ax+by √ a2+b2 )v. Thus T (x, y) = (−bx+ay √ a2+b2 )u − ( ax+by √ a2+b2 )v = ( (−b 2+a2)x−2aby √ a2+b2 , (a 2−b2)y−2abx √ a2+b2 ). Hence the transformation T is linear. Since for null space T (x, y) = O, ( (−b 2+a2)x−2aby √ a2+b2 , (a 2−b2)y−2abx √ a2+b2 ) = O, (0, 0) is the only solution of (x, y). N(T ) = O and nullity is 0. Range is all R and rank = 2.