Two-variable functions play an essential role in multi-variable calculus. They are also essential in different branches of abstract algebra including linear algebra. In this post, first, we discuss two-variable functions from different points of view. Then, we proceed to introduce their role in abstract algebra. A two-variable function is a function whose input is an…

# Author: Dr. Peyman Nasehpour

## Orthogonal subsets

First let us define orthogonal subsets. Let \(O\) be a subset of an inner product real vector space \(V\). By definition, \(O\) is an orthogonal subset of \(V\) if any pair of distinct elements of \(O\) are perpendicular to each other; in other words, if \(O\) does not contain the zero vector and \(u\) and…

## Subspaces of vector spaces

A vector space has many subsets, but some are “smaller vector spaces” called subspaces. More precisely, if \(V\) is a real vector space and \(W \subseteq V\), then it is said that \(W\) is a subspace of \(V\) if \(W\) is itself a real vector space. In the post on linear maps in vector spaces,…

## Cross product

The cross product is one of the most basic and essential operations in the three-dimensional real vector space \(\mathbb{R}^3\). The cross product has significant applications in many fields of science such as physics and computer programming. On the other hand, it is used in many fields of mathematics like differential geometry. Origins of the cross…

## Vector orthogonal projection

Vector orthogonal projection is an essential operation in the Gram-Schmidt orthonormalizing process and the least square problem. In this post, we discuss vector orthogonal projection, give some suitable examples, and solve some exercises. A motivational example of a vector orthogonal projection in analytic geometry Consider the point \(P=(1,7)\) and the line \(L\) passing through the…

## Matrix norms and examples

The main purpose of this post is to introduce different matrix norms such as Frobenius norm and \(p\)-norms. In the following, first we give a general definition of matrix norms. General definition of matrix norms In the post on matrix operations, it is explained that \(M_{n\times n}(\mathbb{R})\) is a real vector space. A matrix norm…

## Norms of vector spaces

Sometimes on a single vector space, different norms can be defined. In this post, first we give the definition of an inner product vector space. Then, we proceed to discuss different norms on the finite-dimensional real vector spaces, i.e. the vector spaces of the form \(\mathbb{R}^n\). The dot product In the post on the dot…

## Linear maps in vector spaces

In this post, we introduce general definition of vector spaces and the linear maps between them. In the post on the finite-dimensional real vectors, we discussed their addition and subtraction, multiple of vectors by numbers, their linear combination, their independence, their dot product, their length and distance, and the angle between them. Real vector spaces…

## Elementary matrix

By definition, a square matrix derived by applying an elementary row operation to the identity matrix is an elementary matrix. Since elementary matrices are very useful tools in matrix computations, we dedicate a post to this kind of special matrices. Elementary matrices have the following interesting property: Let \(E\) be an \(m \times m\) elementary…

## Gaussian elimination

Gaussian elimination is the name of an algorithm which uses a sequence of elementary row operations to solve a system of linear equations. In the post on systems of linear equations, we introduced systems of linear equations, the matrix of coefficients, and the augmented matrix. Here we first introduce the back-substitution method to solve special…