Brunei Math Club

The determinant of a general \(n\times n\) matrix is quite complicated. We give a mechanical, recursive definition first and then think about its meaning later. Definition (Determinant) Let \(A = (a_{ij}) \in M_n\). The determinant \(|A|\) of \(A\) is defined recursively in the following manner. If \(n = 1\), then \(|A| = a_{11}\). If \(n > 1\), then let \(A_{ij}\) denote the \((n-1)\times(n-1)\) matrix obtained by removing the \(i\)-th row and the \(j\)-th column from \(A\), and \[|A| = \sum_{j=1}^{n}(-1)^{i+j}a_{ij}|A_{ij}|\] where \(i\) is any arbitrary index from 1 to \(n\). (Instead of a row, you may use an arbitrary column to obtain the same result.) According to this definition, if we want to compute \(|A|\) of an \(n\times n\) matrix, we need to compute the determinants \(|A_{ij}|\) of (many) \((n-1)\times(n-1)\) matrices, which requires computing determinants of \((n-2)\times(n-2)\) matrices, and so on, until we reach the determinants of \(1\times 1\) matrices which are tri

So far we have been studying the Fourier series of functions with period \(2\pi\). What about the functions with periods other than \(2\pi\)? Furthermore, what about non-periodic functions? In this section, we consider these problems informally and briefly.  Suppose \(f\) is a smooth function with a period of \(2l\pi\) where \(l > 0\) is a constant. We have \[f(x) = \sum_{k = -\infty}^{\infty}c_ke^{ikx/l}\tag{eq:f2lpi}\] and the right-hand side converges uniformly. The Fourier coefficients are given by \[c_k = \frac{1}{2\pi l}\int_{-l\pi}^{l\pi}f(x)e^{-ikx/l}dx ~~~ (k \in \mathbb{Z}).\] Thus, the Fourier series can be readily extended to any periodic function with period other than \(2\pi\). Next, we consider functions that are not necessarily periodic. Suppose that \(f\) is a function of class \(C^1\) with a bounded support.  Remark . The support \(\mathrm{supp}(f)\) of a function \(f\) is the subset of the domain, defined by \[\mathrm{supp}(f) = \{x \in \mathbb{R} \mid f(x) \neq 0