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 $$\begin{array}{}\text{(eq:f2lpi)}& f(x)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_k{e}^{ikx/l}\end{array}$$ 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(...