Previously, we have explored methods to compute the essence of power functions $Ȣx^n$ which involves solving a large system of linear equations. This method is equivalent to solving for the inverse of a large $n\times n$ matrix where entries are values of pascal's triangle. Though the matrix method allow us to solve for large number of essences at once, it does not extend easily to solve for next iterations of essence coefficients. Rather than reusing the values we have already solved for, we will have to solve for inverse of a separate larger matrix again. Here we will introduce an iterative method for solving for these coefficients. Chapter 0: Recap Let us first remind ourselves of the definition of essence. For a function $f(x)$, we want to find the transformation $Ȣf(x)$ such that we are able to 'smooth out' its series: $$\sum_{i=a}^b f(i) = \int_{a-1}^b Ȣf(x) dx$$ For example, we can solve for the following functions: $$\begin{align*}Ȣ1 &= 1 \\ Ȣx &= x +
Chapter 1: the Question How many different types of RREF are possible for a $2\times 3$ Matrix? For example, it can be in $\begin{bmatrix} 1&0&a\\ 0&1&b \end{bmatrix}$ or even $\begin{bmatrix} 1&a&b\\ 0&0&0 \end{bmatrix}$. So what is the total count of RREFs in $2\times 3$? Since we don't think there will be too many, we can try to write them all out: $$ \begin{bmatrix} 1&0&a\\ 0&1&b \end{bmatrix}, \begin{bmatrix} 1&a&0\\ 0&0&1 \end{bmatrix}, \begin{bmatrix} 1&a&b\\ 0&0&0 \end{bmatrix}, \\ \begin{bmatrix} 0&1&0\\ 0&0&1 \end{bmatrix}, \begin{bmatrix} 0&1&a\\ 0&0&0 \end{bmatrix}, \\ \begin{bmatrix} 0&0&1\\ 0&0&0 \end{bmatrix}, \\ \begin{bmatrix} 0&0&0\\ 0&0&0 \end{bmatrix} $$ So in total, we find that there are 7 possible RREFs. But is writing all the possibilities the best approach to this proble