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: $$\begin{align*}Ȣ1 &= 1 \\ Ȣx &= x +
The search for a basis for the orthogonal complement of a subspace has often involved pure guesswork or many tedious computations of removing orthogonal projections from a known basis for the entire vector space. Here we propose an alternative faster method of producing a basis for the orthogonal complement of an image of a matrix. Let $A$ be $n \times m$ matrix over $\mathbb{C}$. Then such matrix maps from $\mathbb{C}^m$ to $\mathbb{C}^n$. $$A:\mathbb{C}^m \rightarrow \mathbb{C}^n$$ (Choice of the field $\mathbb{C}$ is to allow us to use the standard inner product $<\cdot, \cdot>$ to define orthogonality. This, of course, works also for $\mathbb{R}$ ) Then its image $im\,A$ and the orthogonal complement of the image $im\,A ^\perp$ are both subspaces of the codomain $\mathbb{C}^n$ such that $$\mathbb{C}^n = im\, A \oplus im\, A^\perp$$ A Basis for Image of $A$ Our first task will be to determine a basis for the image. I have already discussed this in a separate lengt