Below is a shear transformation matrix A. If you multiply a vector in $\mathbb{R}^{2}$ by the matrix A you will get back a sheared vector in $\mathbb{R}^{2}$.
A = $\begin{bmatrix} 1&2 \\0&1 \end{bmatrix}$
However, the above matrix is also the inverse matrix (A is the inverse of B) of the following:
B = $\begin{bmatrix} 1&-2 \\0& 1 \end{bmatrix}$
If you multiply a vector (standard basis) by the matrix A you will get back the coordinates of that vector relative to the coordinate system defined by matrix B. Correct?
Therefore, a matrix's "meaning" is dependent on the context we use it in. Is that right?