Introduction and roadmap

0:00 ready to learn mathematics behind machine learning generative AI or Cutting Edge teag then you are in the right place today we are going to learn linear algebra like never before in this Crush course in 6 hours we are going to learn the basics and the fundamental concepts in linear algebra a fundamental part in mathematics when it comes to many applied sciences our Crush course in fundamentals to linear algebra will be a great starting point for anyone who wants to refresh their your algebra and knowledge or who wants to start with the fundamentals of mathematics for AI data science machine learning or generative AI so if you are looking for that one course to refresh your memory in those fundamental concepts and be able to create your own algorithms and create your own AI dream products then you are in the right place and if you're really serious about your education and career and backup with your mathematical foundations then you can also check our 20 5 plus our comprehensive fundamental to linear alup course which is also part of our mathematics boot camp so check at lunch.

0:48 and you can get this comprehensive 25 plus hour course along with the certification in linear algebra so here's what we are going to cover as part of our fundamentals to linear algebra crash course so we are going to start with the introduction to linear algebra we're going to talk about the prerequisites what is exactly that you need in order to get into linear algebra whether as a student as part of your undergrad or Graduate Studies or as a working professional who wants to learn the mathematics in a good way in order to become a well-rounded professional whether it is for your future in quantum physics in a future of data science machine learning or artificial intelligence so we are going to learn the basics of the vectors this is the first section that we are going to cover here we are going to learn the vectors their basic definitions the theory but also the examples so in this course we

1:58 are not just going to scratch the surface we are going to dive deep and the idea is that for even for those basic concepts we are going to learn not just the theory the basic definitions but we are also going to implement them in different examples we are going to manually solve different examples by hand in order to truly master and practice this topic and beside this we're also going to talk about the applications of them across different domains including in data science and AI the idea is that those basic concepts are your starting point in the future of linear algebra because to truly understand this topic and to truly understand this entire linear algebra field at least the course the academic level course that you are used to seeing as part of the different Advanced mathematical studies you will need to dedicate more than just 6 hours so if you are serious about your education and career and you want to learn mathematics from ground up then check also our 25 plush hour complete fundamental to linear AR records to truly Master the entire linear algebra in a very economic and efficient way which contains an

3:05 entire semester equivalent material Theory practice and the applications so here's what we are going to cover as part of our crash course so we will start with the introduction to linear algebra we will talk about this concept of linear algebra we will see what are the prerequisites in order to even understand and start with this course we're going to refresh our memory with the different fundamental concepts here in this introduction to linear algebra section we are going to quickly cover the prerequisite that you must know in order to get started with linear algebra here I'm also going to refresh our memory on the fundamental concepts like real numbers and Vector spaces the norm of a vector the cian coordinate system and also the angles triogen ometry the norm versus equid in distance pedian theorem Etc so uh after this we are going to finally Kickstart dealing your algebra Concepts so first up we are going to talk about the basic Vector spaces here we are going to cover the definition of the vectors what are vectors the basic representation of it but also the indexing of it and how we

4:14 can apply different simple operations like vector addition Vector subtraction and also scalar multiplication after this in the next section we are going to talk about the matrices and the basic operations we are going to truly understand the matrices the basic definition of them we are going to see many examples of matrices and also in practice what are the basic operations that we can apply to matrices including the additions the scale and multiplication and subtractions after this we are going to look into the dot product and the vector length in this section we are going to talk about the idea of dot product what is the difference between dot product and inner product understand the fundamentals of that product and how they are calculated we're going to complete many examples as this a very important topic especially for data science machine learning in Ai and once we are done with this we will also look into this concept of the length of a vector and how this is related to for instance cosine rule and these different inequalities when it comes to the vector spaces after to this we are going to look into the basic linear systems how we can solve a linear system by using the infamous gaussian elimination here we are going to talk

5:26 about this basic concept of solving linear equations with a many unknowns using the concept of matrices we're are going to see detailed examples where manually step by step by hand we are going to derve the um entire process using gin elimination how we can solve the system of linear equations first getting the arent Matrix and then this concept of reduce row form so RF and then R RF so row H form and reduce Ro form and uh we are going to look into two different scenarios to ensure that we understand this intuition behind solving this linear systems in various um type of cases now we are going to uh finish off uh the crash course with the preview of the advanced concepts we are going to see the special matrices we are going to see symmetric matrices diagonal matrices and then uh the idea of uh n matrices W matrices how they are used uh and why they are used commonly in programming and then we are going to see a glimpse of how linear algebra can be applied across real world scenarios so in this crash course we are not just going to

6:37 cover the uh quick definitions but we are going to dive deep we are going to truly try to understand those concept those uh geometric intuition and interpretation behind those Concepts and how they are related uh towards many applied sciences so to truly understand how each of those concepts are actually implemented in the real world like in the fields of data science statistics and AI this course is perfectly tailored for working professionals who need to sharpen their math skills who want to uh quickly and efficiently refresh their memory whether is before their interviews or who want to learn machine learning and statistics and AI but they don't have the mathematical background and instead of just going and studying at University like I did and spending many years to do an undergraduate uh studies in mathematics or statistics instead you can just follow this course to quickly refresh your memory but if you truly want to understand all the concepts which is equivalent to one semester or linear algebra course then definitely check out our 25 plus hour fundamentals to linear algebra course as part of our mathematics boot camp at

7:49 l. because you can't master linear algebra in just 6 hours so if you are a student this course will also be a good way for you to refresh your memory if you you are at the end of your deadline and you are about to complete your linear track exam and if you want to learn everything from scratch and you have the time then you can also check our 25 plus hour course to learn the linear algebra fully and to Ace your exams this crash course is just the beginning for you in this way you are laying a solid foundation to become a well-rounded professional and it truly understand this topic of linear algebra so if you're ready I'm really excited so let's let's get started welcome to the mathematics for data science and AI road map in 2024 today we are going to discuss the road map behind a topic that powers many algorithms the linear algebra we are going to talk about all the concepts and the topics that you must know in order to master the linear algebra which is the backbone of data science machine learning generative Ai and many other topics and Cutting Edge Tech whether you are aspiring to become a data scientist or AI professional this

9:01 is your starting point to learn the mathematics of data the fundamentals or linear algebra if you want to become a good professional a well-rounded data scientist and master the art of machine learning generative Ai and many other Cutting Edge Tech domains then your starting point should be to learn the mathematics to truly understand this different mathematical Concepts that power the algorithms behind the scenes including many Infamous algorithms like the GPT series or the bird or Transformers linear algebra is the mathematics of data it's about the vectors matrices projections and how to solve a system of linear equations and how to decompose your Matrix into many parts to simplify the equations and solving of the linear systems think about metric factorization that is used as part of the recommended systems including the ones in the Netflix that uses the streaming company to recommend you movies when you go to the net so linear albra is one of the most fundamental and must to know topics when it comes to mathematics for your data science AI Journey so let's now dive into the road map behind linear algebra

10:14 and how you can learn linear algebra in 2024 I'm going to make use of the curriculum and the road map that we have carefully drafted as part of our 25 plus hour fundamental SAR algebra course which is also part of our mathematics boot camp at lunch. so I'm going to make use of that road map and I'm going to also make use of the applications of lunch. a in order to Showcase you all these different topics as part of linear algebra road map of 2024 so let's di first we need to start with the linear algebra introduction you will need to understand this Concepts behind linear algebra where it is positioned in this entire field of mathematics and why you exactly needing your algebra to study so um first you need to learn the prerequisites and all this must you know topics that you need in order to be able to understand linear algebra think about the real numbers the vector space this concept of the norm of a vector the length of a vector the cation coordinate system which is usually covered as part of the pre-algebra or algebra courses and also the Angles and tri gometry think about the cosine the sign cosine

11:27 rule the sinus rule the Tang function how it looks like when you visualize it the geometric interpretation behind it and also this concept of Al and distance and how Elan distance is different from the norm and finally you also need to understand this concept of Pythagorean theorem and the orthogonality so when it comes to the linear algebra one thing to keep in mind is that linear algebra is usually a course that is part of the undergrad studies like computer science econometrics advanced mathematics advanced statistics and usually the students of the studies they in the second or third year of their bachelor they learn linear algebra and this road map that I'm providing to you is based on academic level plus the industry Insight curriculum which means that uh the um uh concept is quite technical so if you want to truly understand the linear algebra and you are serious about your career and your education then this road map on one hand might seem bit Technical and intimidating but on the other hand you need to understand that students spend about semester on this entire road map but if you understand

12:39 these Concepts this will open many doors for you and instead of just using the libraries of people who have created for example large language models like the gpts or the bird or the uh mixtur uh AI models instead of just using those libraries and pre-trained model you can create your own models and I think that's a good enough motivation for us to truly understand and invest the time and motivation to master the art of linear algebra so once you have these prerequisites that usually come from prealgebra or courses like the Calculus 1 and calculus 2 to uh learn these different concepts you are ready to go onto the first journey and first section behind the linear algebra which is to learn the vectors and operations so be prepared to learn the foundations of the vector s the special vectors and operations and let me actually go ahead and get more uh in detail curriculum for you so um start to understand uh the vectors and operations what are the foundations of vectors the fundamentals of linear algebra the scalers and the vectors this representation of the vectors in terms of the magnitude and

13:49 the direction and also understand this uh Vector notation the indexing of the vectors first upop you can uh start with learning vectors and operations so here you need to understand this uh scalers the vectors the representation of the vectors like the magnitude and Direction what is a common notation of the vectors in the industry as well as this indexing in vectors so understand what the indexing behind vectors are because then you need to implement this in programming languages like python then uh once you are clear on those topics then you can get into the special vectors and operations so understand what are the zero vectors and the unit vectors the sparsity vectors the vectors in higher Dimensions the vector addition and subtraction as well as the uh Vector uh multiplication so scalar multiplication how you can calculate the multiplication behind the scalar and a vector also learn to um the different properties of the vectors and the operations uh of these vectors so once you are done with that get into the advanced Vector Concepts so learn the linear combination concept and try to

15:01 understand this concept of unit vectors the span of a vector what is this definition and the application of it uh the concept of linear Independence is really important understand this concept and also uh be able to uh analyze and see from uh multiple vectors whether they are linearly independent or not look into this concept of scalar multiplication as well as the application of scalar Vector multiplication and look into this concept of a length of a vector and a DOT product as well as what is the difference between the dot product and the inner product after this get into the topic of dot product the coui squares and its application so understand the concept of dot product the inner product what are the differences between the two what is the properties of dot product calculate dot product by hand on many examples and also learn this concept of COI squs CI squs is an Infamous inequality theorem comes from linear algebra and understanding its uh Theory but also intuition maybe you can even derive the proof of it using the cosine law and how that relates to this concept of the norm

16:11 of a vector and when it is that the norm of a vector is equal to zero so this completes then the vector operation section uh in here this is the high level overview of the topics that you must know in order to learn the vectors and the operations and once you are done with this then you need to get into the next section and the uh domain in linear algebra which is the matrices and solving linear systems here I'm talking about foundations of linear systems and matrices the introduction to matrices the theory and the intuition behind it core Matrix operations gaan reduction n space Comm space Rank and the full rank so here think about the uh General linear systems like uh the co efficient labeling homogeneous versus non-homogeneous systems and what is this uh definition of matrices the uh notation of it as well as this this concept of the rows the columns the dimensions of matrices the identity Matrix diagonal matrix other other special type of matrices like uh one's Matrix the zero Matrix but also learn the core Matrix operations and practice

17:24 with them like Matrix operations of addition subtraction scalar multiplication and also the multiplication behind multiple matrices once you are done with this um in terms of the solving of linear systems think about Matrix representation of linear systems the solving of linear systems using the infamous gausian elimination like understanding this argumented coefficient Matrix the row H form the reduced row H form the identity Matrix and the reduced row H form look into the detailed examples implement this in uh practice with multiple examp step by step by um manually deriving the solution to the system as well as how you can calculate a new space a com space and the basis of these different uh spaces on an actual example once you are done with this learn this concept of the rank and in what case you can say that the Matrix has a full rank next up is the section of linear transformation and matrices so learn the algebraic laws of matrices they approves the determinant the concept of the transpose and the inverses of matrices and here this is exactly what I mean in more detail so a way comes to the algebraic

18:36 Row for matrices learn about the cumulative low for matrices associative low distribution low the scaler multiplication low for matrices and look into the proof of this lws as well as the examples after this get into the determinants and their properties look into the uh theory behind determinants why we use determinant how we can calculate related determinant on an example as as what are the properties of determinants and the geometric intuition behind it so when it comes to the Matrix inverses and identity Matrix look into the definition the theory behind it also this non- Singularity of matrices and what is this definition of and the formula when it comes to the inverse of a 2x2 matrix and how you can go from there to a larger matrices let's say 3x3 matrices looking to the detailed examples calculate them inverses of the 2x2 matrices detailed uh in detailed examples and also do the same for the 3X3 matrices so how you can use this idea of conjugates and by changing these different signs to calculate the inverse of a 3X3 Matrix after this get into the

19:49 uh applications of uh inverses of a matrixes why we need them what is the intuition behind it and where do we use it even when we are dealing with the very basic uh machine learning algorithm like um linear regression we use actually the transpose of matrices the inverse of matrices the multiplication behind matrices to find the uh solution to the uh linear system behind the linear regression so uh this is a very important concept that I recommend you to learn before getting into machine learning next up is the transpose of matrices learn the um properties of the transpose of matrices the U theory behind it but also the implementation and the relationship behind the dot product so once you are done with this the next St is to uh get into more advanced section which is the final section way it comes to mastering fundamentals of linear algebra which is this um different Advanced linear algebra topics think about Vector spaces the projections of vectors the grumme process the metrix factorization and look into couple of examples of Matrix factorization techniques that are widely

21:02 used across the industry including the QR de composition the igen de composition which includes the calculation of igon values and igon vectors very popular topic as part of data science and statistical modeling also uh Ai and get into the singular value that composition which is the SVD so let's dive into the detailed uh details of this uh section when it comes to the vector spaces projections and the gred process look into the definition of it the step-by-step process behind it the uh projection formula and the step-by-step example behind the uh uh projection in order to truly understand how you can calculate the projection of a vector and also look into the geometric interpretation and the intuition behind vector projections uh learn the theory but also the implementation of the dotproduct of vectors and the dotproduct of a vector with itself look into the uh relationship between the dot product the vector of ve the projection of vectors and then look into the concept of orthonormal basis understand the orthogonality

22:12 normalization and look into the infamous grme process understand how you can use the IDE of that products but also the projections in order to conduct this grme process step by step once you are done with this then uh look into the application of grme process and the auton normal bases so uh as part of these special matrices and their properties I would recommend you to learn the special matrices like the diagonal matrix the symmetric Matrix The orthogonal Matrix and the different uh applications and examples of them step by step once you are done with this learn this concept of Matrix factorization Matrix factorization is a very important topic when it comes to data science machine learning but also AI uh the metrix factorization different examples and techniques that um are falling under this umbrella of metrix factorization techniques that are used in both in machine learning but also in deep learning and in generative AI so if you want to become a true uh well-rounded professional in this Fields then definitely be prepared to learn metric authorization techniques dive into its applications understand the

23:24 different examples of what kind of metric alization techniques are used in what uh algorithms and this will help you to get more excited and warm up for the next sections which are detailed examples and you truly understand at least couple of metrix factorization techniques so first up uh learn for example the QR de composition which is a matrix factorization technique learn the theory behind it the step-by-step process behind QR de composition how you can calculate the um q's and then the vectors of r so construct The Matrix of Q and then Matrix R and then uh decompose your Matrix a into uh q and then R matrices in order to decompose your entire Matrix look into detailed example the calculation of it uh with a stepbystep manual process and look into its practical consideration and application in solving linear systems after you are done with this look into the concept of IG value the composition how you can calculate the IG values and I vectors what is the theory but also the Practical implications of them where are ion values and igon vectors used for

24:35 example in the infamous principal component analysis or PCA then look into this concept of ion de composition look into detailed examples and manually calculate the ion values and ion vectors as well as the igon value that composition so once you are done with this and you have looked into the applications of igon bi and ion vectors the calculation of it as well as the Theory getting into the final topic which is another Matrix factorization technique an Infamous one which is the singular value de composition the SVD look into the theory behind the SVD the left singular matrices the right singular matrices this uh theory behind them but also the Practical example behind them uh what is this idea behind um singular value de composition how you can construct The Matrix of s The Matrix of v and the Matrix of D and how you can decompose a matrix into the three different matrices by constructing them separately once you are done with this look into detailed example this might take a bit let's say an half hour or 1 hour in the first go to calculate this different matrices by hand but trust me

25:46 it is worth it because you will then truly understand those Concepts and you will understand how and why those uh Matrix factorization techniques are used and applied in the dat science and artificial intelligence welcome to the course on the fundamentals of linear algebra my name is D Vasan and today we are going to start with some basic concepts that are important for understanding linear algebra linear algebra is one of the most applicable areas of mathematics it is used by pure mathematicians that you will see in a universities doing research publishing research papers but also by the mathematically trained scientists of all disciplines this is really one of those areas in mathematics that you will see time and time again appearing in your professional life if you want to become a job ready uh data scientist or you want to do some handson machine learning deep learning Andi stuff but also linear algebra is used in cryptology it is used in cyber security and in many other areas of computer science and artificial intelligence so if you want to become

26:57 this well-rounded Prof professional you want to go beyond using libraries and you want to truly understand the uh mathematics and the technical side of these different machine learning algorithms from very basic ones like linear regression to most complex ones coming from Deep learning like architectures in neural network how the optimization algorithms work how the gradient descent works and all these other uh different methods and models then you are in the right place because you must know linear algebra such that you will understand these different concepts from very basic ones to most advanced ones in data science machine learning deep learning artificial intelligence data analytics but also in many other applied science disciplines so before starting this