A finite-dimensional vector space is a vector space where some actual list (which remember, has finite length) of vectors spans the space.

An infinite-demensional vector space is a vector space that’s not a finite-dimensional vector space.

additional information

every finite-dimensional vector space has a basis

Begin with a spanning list in the finite-dimensional vector space you are working with. Apply the fact that all spanning lists contains a basis of which you are spanning. Therefore, some elements of that list form a basis of the finite-dimensional vector space you are working with. \(\blacksquare\)

Fireside Chats are a group of broadcasts by Franklin D. Roosevelt (FDR) which allowed him to speak directly to the people.

Below you will find a list of the Fireside articles.

First Order ODEs are Differential Equations that only takes one derivative.

Typically, by the nature of how they are modeled, we usually state it in a equation between three things:

\begin{equation} t, y(t), y’(t) \end{equation}

as in—we only take one derivative.

Sometimes the solution may not be analytic, but is well-defined:

\begin{equation} y’ = e^{-x^{2}} \end{equation}

we know that, by the fundamental theorem of calculus, gives us:

\begin{equation} y(x) = \int_{0}^{x} e^{-s{2}} \dd{s} \end{equation}

Consider the case where there are two functions interacting with each other:

\begin{equation} y_1(t) \dots y_{2}(t) \end{equation}

So we have more than one dependent function, with functions \(y_1, y_1’, y_2, y_2’\) and so forth. To deal with this, we simply make it into a matrix system:

\begin{equation} y(t) = \mqty(y_1(t) \\ \dots \\ y_{n}(t)) \end{equation}

For instance, should we have:

\begin{equation} \begin{cases} y_1’ = 3y_1 - 2y_2 \\ y_2’ = -y_1 + 5y_2 \end{cases} \end{equation}