Question

Let T:R³ →R³

denote the linear map corresponding to reflection about the x−y plane. - Find the martrix [T]. Find all vectors
v ∈ R³such that T( v ) = v. Find all vectors v ∈ R³ such that T( v ) = − v
.

Answer 1

The linear map [T] for reflection about the x-y plane in R³ is represented by the matrix [1 0 0; 0 1 0; 0 0 -1]. Vectors v that satisfy T(v) = v lie on the x-y plane with a z-component of 0, while vectors satisfying T(v) = -v are orthogonal to the x-y plane with non-zero z-components.

Step-by-step explanation:

The goal is to find the matrix [T] for the linear map T representing reflection about the x-y plane in R³, and determine all vectors v ∈ R³ such that T(v) = v and T(v) = -v.

The reflection about the x-y plane in R³ implies that points on the x-y plane remain unchanged, while the sign of the z-coordinate is inverted. Thus, the matrix [T] is given by:

• 1 0 0
• 0 1 0
• 0 0 -1

Now, for T(v) = v, the vector v must lie on the x-y plane, which means its z-component is 0. Therefore, all vectors of the form (a, b, 0), where a and b are real numbers, satisfy this condition.

For T(v) = -v, vector v must be orthogonal to the x-y plane, implying its x and y components are zero. Consequently, all vectors of the form (0, 0, c), where c is a non-zero real number, will satisfy this condition.