Change of basis Matrix in a subspace I've been investigating this for quite a long time (also looking at other posts), and I still cannot figure it out. Let's try it.
I do have a subspace $F$ belonging to $\mathbb{R}^{4}$, which is having a base $A=\{\vec{a_{1}}, \vec{a_{2}}, \vec{a_{3}}\}$, each of them having 4 components. There is another base $B$ from the same subspace, and they're asking how the change of basis matrix from B to A would look like. I understand that this matrix should be a 3x3 because:
\begin{equation*}
\vec{a_{1}}=w_{11}\vec{b_{1}}+w_{21}\vec{b_{2}}+w_{31}\vec{b_{3}}\\
\vec{a_{2}}=w_{12}\vec{b_{1}}+w_{22}\vec{b_{2}}+w_{32}\vec{b_{3}}\\
\vec{a_{3}}=w_{13}\vec{b_{1}}+w_{23}\vec{b_{2}}+w_{33}\vec{b_{3}}
\end{equation*}
Then I get the nice change of basis matrix from B to A formed by the different $w_{ii}$, so then I see this $M_{B\rightarrow A}$ is a 3x3 matrix, OK. Then they're giving me specific values for the base A, and also a matrix $M_{B\rightarrow A}$, and the question is, "given the matrix $M_{B\rightarrow A}$ and the specific values for A, provide the coordinates of the basis A in basis B".
My rationale then is to proceed like this: If I have $M_{B\rightarrow A}$ but I want $M_{A\rightarrow B}$ i do $M_{B\rightarrow A}^{-1}$ and then $B=M_{B\rightarrow A}^{-1}\cdot A$. But of course this is not working as $M_{B\rightarrow A}^{-1}$ is 3x3 whilst $A$ is 4x3. And here is where I'm stuck. I know that probably there is a fundamental misunderstanding on the last step. I've done quite a lot of this kind of exercises but not with subspaces but with "complete" spaces.
Any little hint here will be appreciated.
Thanks!
 A: Let's see it this way:
We will call the vectors in $B$ as $b_1, b_2, b_3$.
Coordinates in $B$ means you have a linear decomposition of any vector $v = v_{b_1}b_1+v_{b_2}b_2+v_{b_3}b_3$.
To get coordinates in $A$, you can start with the above and decompose $b_i$ in $A$.
Suppose $b_i = \sum_j M_i^j a_j$. You can see $v$ expressed in $A$ is $\sum_i v_{b_i} \sum_j M_i^j a_j$. Now please write the sums out yourself and look closely how the coefficients $v_{b_i}$ and $M^j_i$ interact. It may help to put arrow signs on the vectors to emphasize abstract vectors. (Coefficients = coordinates. Vectors are abstract.)
The coefficients actually coincide with the matrix-vector multiplication of $[M_i^j][v_{b_i}]$ where I used $[M_i^j]$ to denote the matrix formed by $M_i^j$ etc.
That is why the change of coordinate matrix is formed by combining the coordinates of $\{b_i\}$ in basis $A$ which are 3-dimensional column vectors.
To clarify how dimension works, a vector in the subspace start with 3 coordinates in basis $B$. Hence, the coordinates form a 3-d vector. NOT 4-d!
