# Mathematics - Linear Algebra Introduction

Hello my friends I'm back! I was gone for the weekend and didn't had anything prepared and so here we are again. Today we start a **new subdivision **on this blog. I have a lot more to tell you, but in many things you will need some knowledge of specific mathematical regions. That's why we today start out talking about **Mathematics **as well, starting off with** Linear Algebra**! I will only make a **quick introduction** today and we will see how it goes :) So, without further do let's get started!

# Introduction:

Linear Algebra is the **branch of mathematics that studies vector spaces and linear mappings** between such spaces. So, Linear Algebra starts with vectors and gets even into n-dimensional spaces. But, in our little series we will get into pretty simple stuff. We will first cover **Matrix algebra**, where we will also talk about **methods** that help us **solve linear equations** (getting into **determinants **at some point). Then we will get into **vector spaces**,** linear functions** and lastly **eigenvalues and vectors**. Today, I will only talk about Matrix algebra basics and also give you an introduction to linear systems, where we will use a method that people may remember from highschool!

# Matrix:

A Matrix is an **rectangular array of numbers arranged in rows and columns**. Each array has a **specific size or dimension** that represents the number of rows or columns. A matrix with R rows and C Collumns is a RXC array. So, an** 4x2 array has 4 rows and 2 columns**. **Each item has a specific position that is specified by the row and column index** its on. So, we are talking about** aij items** that are in the** i-row and j-column**, where **i<R** and** j<C** that are the number of rows and columns of the matrix. The items with the same index of i and j constitute the** main diagonal**.

### Array-Vector Types:

- When the number of rows and columns is the same than we are talking about a
**square matrix**. - An array that has only one row is an
**row vector**and an array that has only one column is an**column vector**. - When the array has one row and column and so only one item it's called an
**vector item**. - When the numbers under the main diagonal are all 0 we talk about an
**upper triangular array**, and when the numbers over the main diagonal are 0 we have a**lower triangular array**. - An array that is upper and lower triangular and has numbers different to 0 only in the main diagonal is called a
**diagonal array**! - A
**diagonal array that has only 1'**s is called a**identity matrix**. - When all the numbers are 0 the array is called a
**zero-array**. - When
**inverting the columns with the rows and vise versa**we end up with the**transpose**of an array. - An array with aij=aji is called
**symmetrical**. - When matrix A is equal to the negative transpose of A we are talking about an
**anti-symmetrical array**!

### Operations:

- Two matrixes are
**equal**when**all the items in the corresponding indexes i and j are equal**. - When
**adding**(subtracting) we**add the numbers in the same indexes**and**get a new matrix**as result. - When
**multiplying with a number**(**scalar multiplication)**we simple multiply each number of the array with the number we are multiplying with. - When
**multiplying matrixes**the result is more complicated!**We can multiply 2 matrixes only when the number of columns of the first one is equal to the number of rows of the second one**. For example: A(3x5) * B(5x7) = C(3X7). Each item is equal to a sum:**cij = ai1*b1j + ai2b2j + ... + aik*bkj**.

So, we can have cases like these:

Lastly an important **equation **has to do with the **inverse array** that only **matrixes **that are **inversible **have. When multiplying a Matrix A with another matrix X the result can become a indicator array I, and when that happens X = inverse A or A^-1. So, **A*X = I, when X = A^-1**. This equation will become handy next time when we get into Linear Equation solving using the Gauss method!

# Linear System Example:

Now let's also lastly get into a **simple 2x2 linear systems** that many know how to solve from highschool. Suppose we have x+y =5 and x-y =20.

A Linear System is represented like that:

*x + 2y = 5*

*x - y = 20*

In highschool we learned a method that simply does multiplication on those equations and additions/substractions between those equations.

For example here we could multiply the second one by 2 and then add them so that we get the value of x, and then simply put the value of x in one of the others to get the value of y.

This looks like this:

*x + 2y = 5*

*x - y = 20 (*2)*

*-----------------*

*x + 2y = 5*

*2*x - 2*y = 40*

*+*

*3*x = 45 -> ***x = 15**

*x + 2y = 5 -> 15 + 2y = 5 -> 2y = -10 -> ***y = -5**

* *I hope that you now remember how we solve an linear system and next time we will get into how we solve bigger systems using a matrix method from the mathematician Gauss that uses operations between the rows and columns to form a specific array that then gives us the variable values!

And this is actually it for today! I hope you enjoyed it!

Give me your feedback and get ready for the next mathematic post...Off course I will continue posting programming related stuff, but as I said before in this post, we will need some advanced mathematics later on when we get into more advanced programming stuff and so I thought it would be a great idea to get into a little bit of mathematics beforehand!

Bye!