Suppose $e_{1},\dots ,e_{n}$ is the standard basis and $\underline{f}_{1},\dots,\underline{f}_{n}$ is any basis of $\mathbb{R}^{n}$. Then for all $\underline{w}\in\mathbb{R}^{n}$, there are unique scalars $\lambda_{i}$ and $\mu_{i}$ for which $\begin{array}{rcl}\underline{w}&=&\lambda_1\underline{e}_1+\ldots+\lambda_n\underline{e}_n&\text{and}\\\underline{w}&=&\mu_1\underline{f}_1+\ldots+\mu_n\underline{f}_n\end{array}$ Using matrices we can rewrite these as $Q(\mu_{1},\dots ,\mu_{n})^{T}= (\lambda_{1},\dots,\lambda_{n})^{T}$ where the columns of $Q$ are $\underline{f}_{1},\dots,\underline{f}_{n}$. Then $(\mu_{1},\dots \mu_{n})=Q^{-1} (\lambda_{1},\dots,\lambda_{n})^{T}$. $Q$ is known as the change of basis matrix.