A Linear Map (a.k.a. Linear Transformation) is a function which maps elements between two vector space that follows linear properties.

constituents

• vector spaces \(V\) and \(W\) (they don’t have to be subspaces)
• A function \(T: V \to W\) (when we put something in, it only goes to one place)

requirements

\(T\) is considered a Linear Map if it follows… (properties of “linearity”)

additivity

\begin{equation} T(u+v) = Tu+Tv,\ \forall u,v \in V \end{equation}

homogeneity

\begin{equation} T(\lambda v) = \lambda (Tv),\ \forall \lambda \in \mathbb{F}, v \in V \end{equation}

general form of First-Order Differential Equations

This will depend on both unknown function \(x\), and the independent variable \(t\). These could and could not be separable.

\begin{equation} \dv{x}{t} = F(t,x),\ x(t_{0}) = x_{0} \end{equation}

Let’s imagine \(F\) is “bounded” and “continuous” on \(I \times \omega\), where \(I\) is an open interval about \(t_{0}\) and \(\omega\) is an open subset of \(\mathbb{R}^{n}\), containing \(x_{0}\). \(F\) is bounded; the results are bounded??

functions embedded in vector spaces

We understand that such First-Order Differential Equations will describe a subset of an infinite dimensional vector space.

example: house price prediction

1 dimension

We want to predict sales price from feet above ground.

\begin{equation} h(x) = \theta_{0} + \theta_{1} x \end{equation}

This makes: \(h : \mathbb{R} \to \mathbb{R}\). and the \(\theta = \qty(\theta_{0}, \theta_{1})\) are what we call parameters or weights.

d dimensions

\begin{equation} h(x) = \theta_{0} + \sum_{j=1}^{d}\theta_{j}x_{j} \end{equation}

but this is like clumsy, so if we come up with a special feature \(x_0 = 1\), we can just make it the linear model it is:

Systems of Linear Equations

\begin{equation} T v = v' \end{equation}

every system of linear equations is decomposed into this. Classically, there’s either a unique solution, no solution, infinite solutions—

problems with zero

“zero” is really hard to define. For instance:

\begin{equation} 6.23423 \times 10^{192} - 1 \times 10^{7} = 6.23423 \times 10^{192} \end{equation}

so in this case \(10^{7}\) literally behaves like zero. (small numbers have the opposite problem)

so, we use elementary row operations to make sure that enormous numbers are essentially standardized—if a row has huge numbers, we may want to scale it down to smaller numbers to make them nice.

see linear temporal logic