# Visualizing 3D linear transformations and Gaussian elimination with Python and Manim Matrices are omnipresent in linear algebra. Columns of a matrix describe where the corresponding basis vectors land relative to the initial basis. All transformed vectors are linear combinations of transformed basis vectors which are the columns of the matrix, this is also called linearity. Algorithms that operate on matrices essentially just alter the way vectors get transformed, preserving some properties. Unfortunately, many algorithms are typically presented to students in a numerical fashion, without describing the whole graphical meaning. In this post we will first visualize simple linear transformations and then we will visualize Gaussian Elimination (with row swaps) steps as a sequence of linear transformations. To do this, we will use Python together with a popular Open-Source library manim.

# Using the PA=LU factorization to solve linear systems of equations for many right-hand sides efficiently Linear systems of equations come up in almost any technical discipline. The $PA=LU$ factorization method is a well-known numerical method for solving those types of systems of equations against multiple input vectors. It is also possible to preserve numerical stability by implementing some pivot strategy. In this post we will consider performance and numerical stability issues and try to cope with them by using the $PA = LU$ factorization.

# Measuring the size of a regular language is NP-Hard In this post we will examine an interesting connection between the $\mathcal{NP}$-complete CNF-SAT problem and the problem of computing the amount of words generated by some efficient representation of a formal language. Here, I call a way of representing a formal language efficient, if it has a polynomially-long encoding but can possibly describe an exponentially-large formal language. Examples of such efficient representations are nondeteministic finite automations and regular expressions.

# Computing the lowest common ancestor in a full binary tree The lowest common ancestor (LCA) problem is important in many applications of binary trees. For example, by knowing the lowest common ancestor we can easily compute the shortest path between any two vertices in a tree. The most common way to compute the lca of vertices $u$ and $v$ is to iteratively go up until we get to the root of the subtree containing both $u$ and $v$. This method works only if $u$ and $v$ are on the same level. If not, we can first measure the difference of heights $d$ between $u$ and $v$ and then find the lowest common ancestor of the $d$-th parent of the lowest vertex and the higher vertex.

# Kadane's algorithm and the idea behind it People often refer to Kadane’s algorithm only in the context of the maximum subarray problem. However, the idea behind this algorithm allows one to use it for solving a variety of problems that have something to do with finding a continuous subarray with a given property. The algorithm can also be used in 2d or multidimensional arrays but in this post we will only consider regular one-dimensional arrays.

# Efficient compression of congruence class automations As already discussed in the previous post, any radix-b number $n$ with $n \bmod{m} \in E \subseteq Z$ can be accepted by finite automata in a digit-by-digit manner. However, the construction is not always optimal. The amount of states required is not always the amount of congruence classes. In this post we will examine when exactly the finite automata can be simplified by combining states. Reducing the amount of states will also help produce a much simpler regular expression, for example, with the Kleene construction.

# Numbers in congruence classes are regular languages In this post we will consider natural Radix-b numbers in positional number systems. The congruence class of any such arbitrary natural number can be determined by a finite automata, and thus, intuitively speaking, the language of all Radix-b numbers that satisfy some fixed properties modulo b is regular.

# Extended Euclidean algorithm without stack or recursion Typical implementation of the extended Euclidean algorithm on the internet will just iteratively calculate modulo until 0 is reached. However, sometimes you also need to calculate the linear combination coefficients for the greatest common divisor.

# Call Stack - buffer overflow vulnerability Buffer overflows are a kind of call stack vulnerability that occur when buffers are created on the stack, but accessed improperly. Buffer underruns are typically not so dangerous, because writing in the current stack frame or beyond the stack pointer will only affect local variables on that stack frame. On the other side, buffer overruns can allow the attacker to overwrite the return address and thus even modify the program’s behavior.