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We recalled that, towards the end of Lecture 4, we observed that, if \(A\) is an \(n \times n\) matrix with \(n\) linearly independent rows, then there exists a matrix \(A^{-1}\) such that

\begin{equation*} A A^{-1} = A^{-1}A = I. \end{equation*}

We say that \(A\) is non-singular or invertible. We then noted that multiplying by \(A^{-1}\) (or, indeed, by \(A\)) can be interpreted as a change of basis operation.

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