In this post, we discuss algebra of finite-dimensional vectors and some fundamental concepts in linear algebra. First, we define finite-dimensional vectors. Let \(n\) be a positive integer. An \(n\)-dimensional real vector \(u\) is an ordered \(n\)-tuple (i.e., an ordered sequence with \(n\) elements) of the following form: $$ u = (u_1, \dots, u_n),$$ where \(u_i\)…

# Month: March 2023

## Dot product of vectors

Definition of the dot product in the two-dimensional space The definition of the dot product of two vectors \(u = (u_1,u_2)\) and \(v = (v_1,v_2)\), denoted by \( u \cdot v \), is as follows: $$ u \cdot v = u_1 v_1 + u_2 v_2. $$ Example. Let us compute the dot product of \(u…

## Norm and distance of vectors

In our previous posts, we discussed addition and subtraction of vectors, multiple of a vector by a number, and linear combination of vectors. In this post, we introduce the norm and distance of two-dimensional and three-dimensional vectors. The norm of a vector Let \(u = (a,b) \) be a two-dimensional real vector. One can calculate…

## Linear functions and examples

Since, in linear algebra, linear combination of vectors play an essential role, a function that preserves linearity must be of special importance also. Such functions are called linear functions. Note that in single variable calculus, by a linear function, it is meant a function \( f\) from \( \mathbb{R} \) into \( \mathbb{R} \) defined…

## Linear combination and independence

Linear combination of vectors In this post, we discuss linear combination of vectors and the important concept of linear independence. By definition, a linear combination of two vectors and is a vector of the following form: $$ r u + s v, $$ where and are real numbers. For example, a general linear combination of…

## Multiple of vectors by numbers

The multiple of an arbitrary vector \( u \) by a real number \( r \), denoted by \( ru \), is obtained by multiplying each component of \( u \) by the given number \( r \). In other words, if \( u = (a,b) \) is a two-dimensional vector, then $$ r u…

## Three-dimensional vectors and their addition and subtraction

Three-dimensional vectors are nothing but ordered triples of real numbers. Usually, vectors are denoted by special letters like \( u, v,\) and \( w \) and their components by \( a, b, c, d, e, f \) and so on. For example, the vector \( u \), where its first component is \( r \),…

## Two-dimensional vectors and their addition and subtraction

Two-dimensional vectors are nothing but ordered pairs of real numbers. Usually, vectors are denoted by special letters like \( u, v, \) and \( w \) and their components by \( a, b, c, d, e, f \) and so on. For example, the vector \( u \), where its first component is \( r…

## Convergent real sequences

Convergent real sequences play a crucial role in mathematics for data science. In this post, we will first define real sequences and their convergence. We, then, list some basic properties of convergent real sequences. Finally, we give some examples to clarify the crucial points related to the convergence of real sequences. Convergent real sequences Definition…

## Properties of real numbers

In this post, we discuss the fundamental properties of real numbers. In the post on types of numbers, we briefly discussed different kinds of numbers including real numbers. However, we need to look at real numbers from another perspective. It is because real numbers have unique properties that are important in real analysis. Recall that…