Linear Algebra - 1

Hi,

I have been watching online video lectures about Linear Algebra, and i thought i can make some "notes" about it. So, i will make small notes in multiple blog entries.

The videos are by the Khan Academy(taught by Sal Khan). These videos are just awesome and please share it with everyone you know who might be interested in this.

Link

Of course, i will not make notes for every video. I will just be making some important points.

Here we go:
Now, let's start with basics and define every concept.

So, what is Linear Algebra?

Quoting Wikipedia,

Linear algebra is the branch of mathematics concerning vector spaces, often finite or countably infinite dimensional, as well as linear mappings between such spaces.

 Now, let us go in depth and describe this. 

What is a vector?

A vector can be defined as an ordered set of Real numbers.
So, simply stated, $$V=\{ X_1 , X_2, X_3 ... X_n | X_i \in R \}$$ 

What this means is, a vector is an n-component entity where each of the components belong to the set of real numbers.

For example, consider the 3-Dimensional vector {1, 0, 0}. Now, this is part of the 3-Dimensional space. Now, let us define what a vector space is:

What is a vector space?

A vector space is defined as:$$R^n=\{ (x_1 , x_2, x_3 ... x_n)  |  x_i \in R \}$$ 

This is the set of all "n-component entities" such that for each component, the definition above for a vector holds good. In other words, this is the set of all n-component vectors. The "n" represents the dimensionality of the space.

For example the 3-Dimensional space can be represented as: $$R^3=\{ (x_1 , x_2, x_3 )|  x_i \in R \}$$  
Thus, this set represents all 3-Dimensional vectors.


Now, vectors are entities that are independent of an origin. i.e the vector (1, 0, 0) can be from (0, 0, 0) to (1, 0, 0) or it can be a vector from (2, 0, 0) to (1, 0, 0). They represent an entity that has a magnitude and a direction. 

Vectors can be defined with the following operations:
- Addition
- Subtraction
- Multiplication(Cross and Dot product)


Let us cover some other interesting aspects of  vectors in the next blog entry.
I will try to make sure not to fill one blog entry with too many concepts.