Change of Basis Matrix Satisfies Certain Relation

Question

Let $V$ be a vector space with ordered bases $\beta = \{x_1,...,x_n\}$ and $\beta' = \{y_1,...,y_n\}$. If $Q = [I_V]_{\beta'}^{\beta}$, then $Q$ is the matrix that changes $\beta'$-coordinates to $\beta$-coordinates, and I am told $Q$ satisfies $y_j = \sum_{i=1}^n Q_{ij}x_i$. However, I am having trouble seeing why this is the case. Would someone help spell out the details?

Answer 1

It's more or less the definition of the matrix of a linear transformation. For a linear operator $T\in\mathcal{L}(V)$, we have that the $j$th column is given as $Ty_j$. Since $\beta$ is also a basis, we can express $Ty_j$ as a linear combination $Ty_j=\sum_iQ_{ij}x_i$. Now in your case, $T$ is just the identity operator, so $Ty_j=y_j$.