Trying to understand change of basis

Question

I am trying to understand the solution to the following problem. In particular, I'm having trouble understanding the first two sentences of the solution ("We know that [...] which we are given"). Could someone please elaborate on the chain of logic?

Answer 1

Sure. The change of basis matrix from $\mathcal E$ to $\mathcal B$ is the first matrix, whose inverse is given. That's because when applied to the basis $\mathcal B$, expressed in terms of itself, you get the elements of $\mathcal B$ expressed in terms of $\mathcal E$.

But we are given that this matrix is invertible. That is, we are given its inverse.

Go from there.

One more thing: the terminology "coordinate isomorphism" is being used for the transformation whose matrix is the inverse of the change of basis matrix. This makes sense, because it takes vectors expressed in one basis, and gives them back expressed in the other. So, $S_{\mathcal B\to\mathcal E}$ (the matrix of $L_{\mathcal E}$) changes coordinates from $\mathcal B$ to $\mathcal E$ (and would normally be called the change of basis matrix from $\mathcal E$ to $\mathcal B$). The reason for this is the well-known conversion formula: $[M]_{\mathcal B}=S_{\mathcal B\to\mathcal E}^{-1}[M]_{\mathcal E}S_{\mathcal B\to\mathcal E}$.

The terms contravariant and covariant are used to express this state of affairs.