Let $$f : \Bbb R^3 \to \Bbb R^4; $$ $$ (a,b,c) \mapsto (a,b,c,a+b+c) $$ Furthermore let $A =\{v_1,v_2,v_3\},B =\{w_1,w_2,w_3,w_4\}$ be bases of $ \Bbb R^3$ and $\Bbb R^4$ respectively where
$v_1=(3,4,4),v_2=(1,1,1),v_3=(0,-7,6)$
$w_1=(2,3,2,-1),w_2=(1,-2,3,-4),w_3=(0,0,0,1),w_4=(-3,8,7,7)$
Find the matrix representation $M^B_A(f)$.
Okay, What I have to do is:
1) Find $M_A^K(f)$ Where K is the standard basis.
2) Put all the column vectors of $M_A^K(f)$ as a linear combination of the vectors in B.
3) Put the coefficients for that linear combination in a new matrix.
I am having trouble with part 2.
$M_A^K = \begin{pmatrix} 3 & 1 & 0 \\ 4 & 1 & -7 \\ 4 & 1 & 6 \\ 11 & 3 & -1 \\ \end{pmatrix}$
since $f(3,4,4) = (3,4,4,11)$ and so on. Now I must use all the column vectors here to find a linear combination for each one of the the vectors in B. Already for the first one I run into trouble given that
$$ \left[ \begin{array}{ccc|c} 3 & 1 & 0 & 2 \\ 4 & 1 & -7 & 3 \\ 4 & 1 & 6 & 2 \\ 11 & 3 & -1 & -1 \\ \end{array} \right] $$ has no solutions. I don't know what I missed. I checked and $v_i$ and $w_i$ are indeed linearly independent so it's not that. I should also add that I made the exercise up so there could be something intrinsically wrong with the question.
What did I do wrong?