Solving maths problems is much more easier than develope an intuition about those problems. YouTube channel 3BLUE1BROWN can help you developing intuition about some really important topics of Mathematics such as linear algebra, calculus, Fourier transform etc. I have been watching videos of this channel for couple of years now and I'm really a fan of this guy-
His name is grant Sanderson who has created this YouTube channel.
Today I'm going to review his videos on linear algebra series. I can not review all the videos of this series in this blog because i don't want you to get bored reading this blog. I'll review only first two videos of this series and I'll review other videos in next blogs.
So, let's start with the first video
Here is the link for the first video
This is an introductory video on vectors where vectors are introduced with 3 different perspectives. As a Mathematics student what I know is, Mathematicians have many more mental images for vectors than physicists. Physics students see a vector as an arrow with magnitude and direction property in some Euclidean space while for a mathematician, vectors are much more general entities.
In LINEAR ALGEBRA, there are two different perspectives to see a vector:
1) geometric perspective
2) algebraic perspective
Both the perspective are equally important
I think this is a nice way to start linear algebra.
In most of the books on linear algebra, you won't find such introduction. They start by defining determinants which doesn't define-
What does it really mean by a determinant?
These questions can be frustrating sometimes and you won't be able to get all answers unless you finish whole chapter on linear algebra then start thinking and linking various terms and their meanings. So, I think vector is really a good starting point of linear algebra.
In LINEAR ALGEBRA
Algebraically, a vector is a set of numbers written in column and if you have studied matrices before, you can see this is just a column matrix where the element in 1st row is horizontal component of the vector and element in 2nd row is the vertical component of the vector.
Honestly, when I first studied matrices I did not have any idea about what am I going to do with a matrix. What I knew was just various rules and methods to solve problems. I couldn't even imagine a relationship between geometry and matrices.
He also discussed about two very important (and also sufficient) operations namely :
1)addition, and 2) scalar multiplication
These two operations are sufficient to build whole set of vectors by using only two vectors (for 2 dimensions). These 2 vectors are called linearly independent vectors (this you'll see in his next video)
Once you'll understand such concepts of addition and scaling of vectors in 2 dimensions, you can generalise these concepts for higher dimensions.
So, this first video is really amazing in terms of visual understanding of concepts such as addition and scalar multiplication and to bridge the gap between algebraic stuff and geometric stuff of LINEAR ALGEBRA.
Let's take the second video
In this video he discussed about, 1) linearly dependent and linearly independent vectors and
2) basis vectors, and 3)span
If you are a high school student then these
terms could be intimidating for you and probably you won't understand what they mean (This is my experience. I didn't get them at the first time when i was taught these things in the classroom). If you know little bit about these terms then this video is surely for you to get more understanding about them.
Lemme share my experience about what I understood when I was taught linearly dependent and independent vectors for the first time.
I used to get confused about linearly independent vectors. I thought only two orthogonal vectors could be linearly independent because if you take any two arbitrary vectors(not orthogonal), I can take component of one of them in the direction of other. This means component of one vector is just the scalled version of the other. This should make them dependent to each other. Fortunately, I understood soon how wrong i was.
This video is surely for you if you have the same confusion about linearly independent vectors.
When this confusion vanishes, you'll understand we can choose any two vectors (as long as we discuss about 2 dimensions) as basis vectors and the set of all linear combination of basis vectors is called their span.
I WOULD HIGHLY RECOMMEND THIS YOUTUBE CHANNEL FOR ALL THOSE WHO ARE INTERESTED IN MATHEMATICS AND WANTS TO KNOW THE DEEP MEANING BEHIND VARIOUS TOPICS OF MATHEMATICS.
I'LL ALSO RECOMMEND TO CHECK HIS VIDEO ON FOURIER TRANSFORM. THIS WILL BLOW YOUR MIND. WHAT YOU MAY HAVE LEARNT, IN THE CLASSROOM OR FROM THE BOOKS, ABOUT FOURIER TRANSFROM WOULD BE COMPLETELY DIFFERENT FROM WHAT YOU CAN UNDERSTAND WATCHING THIS VIDEO. I'LL BET YOU WON'T REGRET WATCHING THIS VIDEO. HERE'S THE LINK
https://youtu.be/spUNpyF58BY
At the last I would like to thank everyone reading my blog so patiently.
Creative
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