gravity sucks.

general relativity claims that our best theory of how gravity work does not work with non-

Each process is controlled by a struct which contain information about the process.

• memory used by the process
• file descriptor table
• thread state
• other accounting

file descriptor table

Within each process, we have a file descriptor table (and the ints we get are indicies into this table), for which each entry stores points to the open file table.

When a process forks, the child doesn’t get more open file entries, instead, we simply clone the file descriptor table (i.e. parent and child will share the same underlying open file table entries); this is how we can share pipes.

Take two linear maps \(T \in \mathcal{L}(U,V)\) and \(S \in \mathcal{L}(V,W)\), then \(ST \in \mathcal{L}(U,W)\) is defined by:

\begin{equation} (ST)(u) = S(Tu) \end{equation}

Indeed the “product” of Linear Maps is just function composition. Of course, \(ST\) is defined only when \(T\) maps to something in the domain of \(S\).

The following there properties hold on linear-map products (note that commutativity isn’t one of them!):

associativity

\begin{equation} (T_1T_2)T_3 = T_1(T_2T_3) \end{equation}

identity

\begin{equation} TI = IT = T \end{equation}

A product of vector spaces is a vector space formed by putting an element from each space into an element of the vector.

constituents

Suppose \(V_1 \dots V_{m}\) are vector spaces over the same field \(\mathbb{F}\)

requirements

Product between \(V_1 \dots V_{m}\) is defined:

\begin{equation} V_1 \times \dots \times V_{m} = \{(v_1, \dots, v_{m}): v_1 \in V_1 \dots v_{m} \in V_{m}\} \end{equation}

“chain an element from each space into another vector”

additional information

operations on Product of Vector Spaces

The operations on the product of vector spaces are defined in the usual way.

Let \(U_1, \dots, U_{m}\) be subspaces of \(V\); we define a linear

We define \(\Gamma\) to be a map \(U_1 \times \dots U_{m} \to U_1 + \dots + U_{m}\) such that:

\begin{equation} \Gamma (u_1, \dots, u_{m}) = u_1 + \dots + u_{m} \end{equation}

Essentially, \(\Gamma\) is the sum operation of the elements of the tuple made by the Product of Vector Spaces.

\(U_1 + \dots + U_{m}\) is a direct sum IFF \(\Gamma\) is injective

Proof: