Matrix function from canonical base to base $B$ Let's say I have a function $F=\begin{bmatrix}\frac{1}{2} & 1 & \frac{1}{2}\\ 0 & 1 & 0\\ -\frac{1}{2} & 1 & \frac{3}{2}\end{bmatrix}$ defined with respect to the canonical basis. 
I found out that the respective function defined with three variables is: $$f(x,y,z)=(\frac{1}{2}x+y+\frac{1}{2}z,y, -\frac{1}{2}x+y+\frac{3}{2}z)$$ 
Now, I would like to define the function $F$ with respect to the basis $B=\{(1,0,1), (1,1,1), (1, 1, -1) \}$ 
So I did:
$$f(1, 0, 1)_{|B}=(1, 0, 1)_{|B}, f(1, 1, 1)_{|B}=(2, 1, 2)_{|B}, f(1, 1, -1)_{|B}=(1, 1, -1)_{|B}$$
My textbook states that the matrix $F_{|B} = \begin{bmatrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$, but shouldn't it be $F_{|B} = \begin{bmatrix}1 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 2 & -1\end{bmatrix}$?

Can anyone look at my question? 
Find basis with given vector representation
I don't have enough reputation to put correct tags.
