Find the matrix of a linear transformation3. Let $f$$ \begin{pmatrix}
        \begin{pmatrix}
        x_1  \\
        x_2  \\
        x_3  \\
        \end{pmatrix}
 \end{pmatrix}
$=
$        \begin{pmatrix}
        x_1+2x_2+x_3  \\
        x_1-2x_2  \\
        x_1+x_3  \\
         3x_1-4x_2          \\
        \end{pmatrix}
$ be a linear map from $\Bbb{R}^3 \to \Bbb{R}^4$
Let $v_1=        \begin{pmatrix}
       1  \\
       0   \\
        1  \\
        \end{pmatrix}
$ $v_2=        \begin{pmatrix}
       0  \\
       1   \\
        0  \\
        \end{pmatrix}
$ $v_3=        \begin{pmatrix}
       1  \\
       0   \\
        0  \\
        \end{pmatrix}
$ and $w_1=        \begin{pmatrix}
       1  \\
       0   \\
        1  \\
0\\
        \end{pmatrix}
$ $w_2=        \begin{pmatrix}
       0 \\
       1   \\
        0  \\
1\\
        \end{pmatrix}
$ 
$w_3=        \begin{pmatrix}
       2  \\
       0   \\
        0  \\
0\\
        \end{pmatrix}
$ 
$w_4=        \begin{pmatrix}
       0  \\
       0   \\
        0  \\
2\\
        \end{pmatrix}
$ Let $B=(v_1,v_2,v_3)$ and $ C=(w_1,w_2,w_3,w_4)$
Find the matrix $M_{B,C}(f)$
Hi so this is the question . What i did was i took a vector $v_1$ i applied f to it and got a vector like $(2,1,2,3)^T$ then i tried to find the coefficeients of the linear combination from $f(v_1)=aw_1+bw_2+cw_3+dw_4$ and solved the system of equation. I put the solutions as coloumns in my matrix. I repeated this 3 times and got a matrix like
$        \begin{pmatrix}
       2  & 0 &  1  \\
       1 &-2 & 1   \\
        1  &-1 &0 \\
     2 &-1 &1\\
        \end{pmatrix}
$ 
My question is how can i check if what i did was correct ?

Here is another way to do it:
$B$ is the representation of the principle vectors of B in the standard basis.
$B^{-1}$ is then the representation of the principle vectors in the standard basis as represented in the basis of $B.$
We can do a similar exercise for $C.$
$fB$ would take the principle vectors in $B$ to a matrix in the standard basis of $\mathbb R^4.$  
Then we have to convert a matrix in the standard basis into the basis for $C.$
$C^{-1}fB$
