First order IVP

The class of problems described as:

\begin{equation} \dv{y}{t} = f(t, y) \end{equation}

and:

\begin{equation} y(t_0) = y_0 \end{equation}

we need to figure “which of the general solutions of the DiffEqu satisfy the general value.

To do this, we simply have to plug in the initial value and solve for our constant \(K\).

Second order IVP

\begin{equation} \dv[2]{d}{t} = f(t,y,y’) \end{equation}

this requires two initial conditions to fully specify (because two variables becomes constant and goes away).

An injective function is one which is one-to-one: that it maps distinct inputs to distinct outputs.

constituents

• A function \(T: V \to W\)

requirements

\(T\) is injective if \(Tu = Tv\) implies \(u=v\).

additional information

injectivity implies that null space is \(\{0\}\)

Proof: let \(T \in \mathcal{L}(V,W)\); \(T\) is injective IFF \(null\ T = \{0\}\).

given injectivity

Suppose \(T\) is injective.

Now, we know that \(0\), because it indeed gets mapped by \(T\) to \(0\), is in the null space of \(T\).

constituents

• \(V\) a vector space
• \((u,v)\), an ordered pair of vectors in \(V\) (its not commutative!)

requirements

We define \(\langle u, v \rangle \in \mathbb{F}\) as the inner product of \((u,v)\) in that order!. It carries the following properties:

1. positivity: \(\langle v, v\rangle \geq 0, \forall v \in V\)
2. definiteness: \(\langle v, v\rangle = 0\) IFF \(v = 0\)
3. additivity in the first slot: \(\langle u+v, w\rangle = \langle u, w \rangle + \langle v, w \rangle\)
4. homogeneity in the first slot: \(\langle \lambda u, v \rangle = \lambda \langle u, v \rangle\)
5. conjugate symmetry: \(\langle u,v \rangle = \overline{\langle v,u \rangle}\)

additional information

Inner Product Space

An Inner Product Space is a vector space with a well-defined inner product. For instance, \(\mathbb{F}^{n}\) has the canonical inner product named Euclidean Inner Product (see below, a.k.a. dot product for reals). The existence of such a well-defined inner product makes \(\mathbb{F}^{n}\) an Inner Product Space.

an integer (\(\mathbb{Z}\)) is the natural numbers, zero, and negative numbers: …,-4,-3,-2,-1,0,1,2,2,3

representing integers

• what are the limitations of computational arithmetic
• how to perform efficient arithmetic
• how to encode data more compactly and efficiently

See also computer number system