F^n — Jemoka Knowledge Base

\mathbb{F}^n is the set of all lists of length n with elements of \mathbb{F}. These are a special case of matricies. Formally—

\begin{equation} \mathbb{F}^n = \{(x1,\ldots,x_n):x_j\in\mathbb{F}, \forall j =1,\ldots,n\} \end{equation}

For some (x_1,\ldots,x_n) \in \mathbb{F}^n and j \in \{1,\ldots,n\}, we say x_j is the j^{th} coordinate in (x_1,\ldots,x_n). additional information addition in \mathbb{F}^n Addition is defined by adding corresponding coordinates:

\begin{equation} (x1,\ldots,x_n) + (y_1,\ldots,y_n) = (x_1+y_1, \ldots,x_n+y_n) \end{equation}

addition in \mathbb{F}^n is commutative If we have x,y\in \mathbb{F}^n, then x+y = y+x. The proof of this holds because of how addition works and the fact that you can pairwise commute addition in \mathbb{F}.

\begin{align} x+y &= (x_1,\ldots,x_n) + (y_1,\ldots,y_n)\\ &= (x_1+y_1,\ldots,x_n+y_n)\\ &= (y_1+x_1,\ldots,y_n+x_n)\\ &= (y_1,\ldots,y_n) + (x_1,\ldots,x_n)\\ &= y+x \end{align}

This is a lesson is why avoiding explicit coordinates is good. additive inverse of \mathbb{F}^n For x \in \mathbb{F}^n, the additive inverse of x, written as -x is the vector -x\in \mathbb{F}^n such that:

\begin{equation} x+(-x) = 0 \end{equation}

Which really means that its the additive inverse of each of the coordinates. scalar multiplication in \mathbb{F}^n At present, we are only going to concern ourselves with the product of a number \lambda and a vector \mathbb{F}^n. This is done by multiplying each coordinate of the vector by \lambda.

\begin{equation} \lambda (x_1,\ldots,x_n) = (\lambda x_1, \lambda, \lambda x_n) \end{equation}

where, \lambda \in \mathbb{F}, and (x_1,\ldots,x_n) \in \mathbb{F}^n. The geometric interpretation of this is a scaling operation of vectors.