Linear transformation of matrices

[geft]

Sorry for the poor formatting, I keep getting errors with Latex in preview. I'll appreciate anyone who could reformat the question into Latex.

Let T: R³ -> R² be defined by T(x, y, z)^T = (x +z, y - z)^T. Find the matrix of T with respect to the bases B = {(1,1,0)^T,(0,1,1)^T,(1,0,1)^T} and B' = {(1,1)^T,(-1,1)^T}.

Here is what I've done. I don't know what's wrong and I'm going crazy because I can't find what's wrong with it. I believe it's my careless mistake, but I've been checking again and again to no avail.

T(1,0,0)^T = (1, 0)^T
T(0,1,0)^T = (0, 1)^T
T(0,0,1)^T = (0, -1)^T

A_T = (1 0 0, 0 1 -1)^T
P_B = (1 0 1, 1 1 0, 0 1 1)^T
P_B' = (1 -1, 1 1)^T

P_B'^-1 = 1/2 (1 1, -1 1)^T

P_B'^-1A_TP_B = (1 0 0, 0 0 -1)

T(u₁) = T(1,1,0)^T = (1, 1)^T $$\neq$$ v₁ + 0v₂
T(u₂) = T(0,1,1)^T = (1, 0)^T $$\neq$$ 0v₁ + 0v₂
T(u₃) = T(1,0,1)^T = (2, -1)^T $$\neq$$ 0v₁ + -v₂

 

[mighty2000]

Let T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 be defined by T(\textbf{x}) = (x_1 + x_3, x_2 - x_3)^T. Find the matrix of T with respect to the bases B = \{(1,1,0)^T,(0,1,1)^T,(1,0,1)^T\} and B' = \{(1,1)^T,(-1,1)^T\}.

Here is what I've done. I don't know what's wrong and I'm going crazy because I can't find what's wrong with it. I believe it's my careless mistake, but I've been checking again and again to no avail.

T(\textbf{e}_1) = T(1,0,0)^T = (1, 0)^T
T(\textbf{e}_2) = T(0,1,0)^T = (0, 1)^T
T(\textbf{e}_3) = T(0,0,1)^T = (0, -1)^T

A_T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -1 \end{pmatrix}
P_B = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix}
P_{B'} = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}

P_{B'}^{-1} = \frac{1}{2} \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}

P_{B'}^{-1}A_TP_B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}

T(\textbf{u}_1) = T(1,1,0)^T = (2, 1)^T \neq v_1 + 0v_2
T(\textbf{u}_2) = T(0,1,1)^T = (1, 0)^T \neq 0v_1