To find the matrix representation of the linear mapping T: R³ → R³, we need to determine how the linear mapping T transforms the basis vectors of R³.

Let's denote the basis vectors of R³ as {e₁, e₂, e₃}, and the basis B as {(1,0,1), (-1,2,1), (2,1,1)}.

1. Finding the Image of Basis Vectors:
First, we need to determine the image of each basis vector under the linear mapping T. We apply the linear mapping T to each basis vector to obtain the images.

T(e₁) = T(1,0,0) = (0,0,0)
T(e₂) = T(0,1,0) = (0,0,1)
T(e₃) = T(0,0,1) = (1,0,0)

2. Writing the Image Vectors in terms of Basis B:
Next, we express the images obtained in the previous step in terms of the basis B. We find the coefficients of the basis vectors B that can be multiplied with the basis vectors to obtain the image vectors.

(0,0,0) = a₁(1,0,1) + a₂(-1,2,1) + a₃(2,1,1)
(0,0,1) = b₁(1,0,1) + b₂(-1,2,1) + b₃(2,1,1)
(1,0,0) = c₁(1,0,1) + c₂(-1,2,1) + c₃(2,1,1)

Solving these systems of equations, we find the coefficients:
a₁ = 0, a₂ = 1, a₃ = 0
b₁ = 1, b₂ = 0, b₃ = 0
c₁ = 0, c₂ = 0, c₃ = 1

Therefore, we can write the images in terms of the basis B as:
T(e₁) = 1(-1,2,1) + 0(2,1,1) = (-1,2,1)
T(e₂) = 0(1,0,1) + 0(-1,2,1) + 1(2,1,1) = (2,1,1)
T(e₃) = 0(1,0,1) + 0(-1,2,1) + 1(2,1,1) = (2,1,1)

3. Constructing the Matrix Representation:
Now, we can construct the matrix representation of the linear mapping T by arranging the images of the basis vectors as columns.

The matrix representation of T relative to the basis B is:
[ -1 2 2 ]
[ 2 1 1 ]
[ 1 1 1 ]

This matrix represents the linear mapping T: R³ → R³, where T(x,y,z) = (z,y,z), relative to the basis B.