Let T R3 rightarrow R3 be the linear transformation given by

Let T: R^3 rightarrow R^3 be the linear transformation given by reflecting across the plane -x_1 + x_2 + x_3 = 0. a. Find an orthogonal basis {v_1, v_2, v_3} for R^3 so that v_1, v_2 span the plane and v_3 is orthogonal to it. b. Give the matrix representing T with respect to your basis in a. c. Use the change-of-basis theorem to give the matrix representing T with respect to the standard basis.

Solution

(a) orthogonal basis is given by:

given v₁ and v₂ span along the plane and v₃ is orthogonal to the plane hence let

v₁ = (1 1 0) , v₂ = (1 0 1) and v₃ = (0 0 1)

T=(-1 1 1)

check for the orthogonolity

(1 1 0).(-1 1 1) = -1+1+0 = 0

(1 0 1).(-1 1 1) = -1+0+1 = 0

(0 0 1).(-1 1 1) = 0+0+1 =1

hence the orthogonal basis is v₁ = (1 1 0) , v₂ = (1 0 1) and v₃ = (0 0 1)(

(b) Given with respect to the basis in a matrix T is represented as (-1 1 1)