The average rate of change of a function can be calculated by dividing the difference between the dependent variable and the independent variable. In this post, we discuss how to compute the average rate of change of functions in different spaces. The average rate of change in single variable calculus Let \(y = f(x)\) be…

# Category: Multi-Variable Calculus

## The inverse of a matrix

The multiplicative inverse of a matrix is a matrix that when multiplied by the original matrix from both sides yields the identity matrix. In other words, the multiplicative inverse of a matrix \(A\) is a matrix \(B\) such that $$AB = BA = I,$$ where \(I\) is the identity matrix. A matrix \(A\) is invertible…

## 1-to-1 and onto linear maps

Linear maps are essential in linear algebra, and 1-to-1 and onto linear maps are especially important. In this post, we investigate 1-to-1 and onto linear maps and discuss them with suitable examples. 1-to-1 linear maps A function is, by definition, 1-to-1 (also called one-to-one or injective) if \(f(x) = f(y)\) implies \(x=y\), for all \(x\)…

## Orthogonal projection

In the post on vector orthogonal projection, we discussed the concept of orthogonal projection of a vector onto a nonzero vector. In this post, we will discuss the concept of orthogonal projection of a vector onto a nonzero subspace of an inner product vector space. The orthogonal decomposition theorem First, we prove the following result:…

## Orthogonal complement

The orthogonal complement of a subset \(W\) of an inner product real vector space \(V\) is the set of all vectors \(u\) in \(V\) such that \(u\) is orthogonal to each element of \(W\). An example of an orthogonal complement As a starting point for our discussion, let us look at the following interesting example:…

## Basis for vector spaces

In this post, we introduce the fundamental concept of the basis for vector spaces. A basis for a real vector space is a linearly independent subset of the vector space which also spans it. More precisely, by definition, a subset \(B\) of a real vector space \(V\) is said to be a basis if each…

## Two-variable functions

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…

## 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…