Vector space engineering mathematics Sc. Rank-Nullity theorem (without Share your videos with friends, family, and the world Share your videos with friends, family, and the world For notes and material join Apka apna telegram group: https://t. It contains well written, well thought and well explained computer science and programming articles, quizzes and De nition: A vector space consists of a set V (elements of V are called vec-tors), a eld F (elements of F are called scalars), and two operations An operation called vector addition that takes two vectors v;w2V, and produces a third vector, written v+ w2V. Eigenvalues and Eigen vectors. with vector spaces. i. youtube. E. 1 Let V be a vector space, and 1 and 2 in V be such that x+ 1 = x and x+ 2 = x 8x2X: Then 1 = 2. In Z the only addition is 0 C0 D0. Definition \(\PageIndex{1}\): Vector Space A vector space \(V\) is a set of vectors with two operations defined, addition and scalar multiplication, which satisfy the axioms of addition and scalar multiplication. MODULE – III. Using the hypothesis and axioms (a) and (c), we have 2 = 2 + 1 = 1 + 2 = 1: Finite dimensional vector space, linear dependent and independent, related theorem 8 9. Statistics Solved Example of Application of Vector Calculus in Engineering Mathematics. , α ∈ V, β ∈ V α + β ∈ V (ii) Associative: α + (β + α + β) + α, β, ∈ V (iii) Existence of identity: an A vector space or a linear space is a group of objects called vectors, added collectively and multiplied (“scaled”) by numbers, called scalars. A vector space W is said to be a subspace of a vector space V if it satisfies the following conditions: W \(\neq\) \(\emptyset\) systems of linear equations, vector spaces and linear transformations, eigenvalue problems, and other topics. Vector Space There is very little chance that you will ﬂip to a page that does not have at least one vector space on it. We have provided Mathematics 1st Year Study Materials and 1. Kreyszig : ³Advanced Engineering This concept is explored in this section, where the linear transformation now maps from one arbitrary vector space to another. sets are subsets of vector spaces or linear mapping is function from a vector space into vector space. We often need to derive a conclusion from Any representation of a plane curve or space curve using a vector-valued function is called a vector parameterization of the curve. This Playlist of Vector Space is helpful for students preparing for B. Assignment-Problems: Vector spaces. It allows us to perform all operation on Vector Space Theory A course for second year students by Robert Howlett typesetting by TEX. 2 Cylindrical Coordinates 5. In this course you will be expected to learn several things about vector Vector Spaces; Basis and Dimension of a Vector Space. Mastering Vector Space in Linear Algebra Doc | 37:58 min: Vector Space Question - 1 Doc | 07:59 min: Vector Space Question - 2 Doc | 03:40 min: Vector Explore Further: Skills Needed to Succeed in Aerospace Engineering. Tech (Engineering) & other competitive exams like IIT-JAM, CSIR NET, GATE. E: Exercises This page titled 9: Vector Spaces is shared under a CC BY 4. 3: Subspaces; 4. . A vector addition MATHEMATICS FOR AEROSPACE ENGINEERS L-T-P-D: 3-0-0-0 Units: 4 Course Instructor: Dr. Prove that the Subset 𝑊 = {(𝑥, 𝑦, 𝑧) \𝑥 − 3𝑦 + 4𝑧 = 0} of the vector space R 3 is a subspace of R 3. P. (1. For any pair of elements ~x;~y2Vthe vector sum ~x+ ~ybelongs to V. Apostol, Wiley This quiz covers questions from Unit-1 of the Engineering Mathematics-I curriculum for the first semester of 2024-2025 at Karnavati University. 3 Further Properties of a Vector Space 3. Linear 5. y cs Contents Chapter 1: Preliminaries 1 §1a Logic and common sense 1 §1b Sets and functions 3 §1c Relations 7 ical nature, being just as relevant to other parts of mathematics as they are to vector space theory. Let V be a non empty set whose elements are vectors. The whole book has been divided into topicwise sections. 8 Summary 3. 1: Deﬁnition of vector spaces; 4. These videos can watch all graduat SOLUTION: Let the equation of the best fit line be of the form 𝒃= + 𝒕. Dimension of a vector space: PDF unavailable: 14: 13. 1 Cartesian Coordinates 5. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes. This number, which is shared by all bases and expresses the number of “degrees of freedom” of the space, is the dimensionof 𝑽. Muhammad Khalid of Early in the development of mathematics, the concept of a vector led to a variety of applications in physics and engineering. INTENDED AUDIENCE : All branches of science and engineering 1 Vector Space A vector space is de ned as a set of vectors V and the real numbers R (called scalars) with the following operations de ned: Vector Addition: V V !V, represented as u+ v, where u;v 2V. pdf), Text File (. , M. Scalars are often real numbers, but can also be complex numbers. Do check the meme at the end, but only after studying 4 Mathematics for Engineering Let be a scalar field in this space and a vector field. Zero Vector: The zero vector Full syllabus notes, lecture and questions for Lecture 6 - Linear Transformations - Linear Algebra - Engineering Mathematics - Engineering Mathematics - Plus excerises question with solution to help you revise complete syllabus for Linear Algebra - Best notes, free PDF download. 2 Vectors and Matrices It is easiest to think of a vector as a generalisation of a single number. See examples of matrices, functions, and solutions as vectors in different We shall de ne the mathematical object, called a \vector space\ which is a composite object consisting of a eld of \scalars", a set of \vectors", and two operations (addition and scalar Vector space is a fundamental subject in mathematics. As an area of study it has a broad appeal in that it has many applications in engineering, physics, geometry, computer science, economics, and other areas. For example, a vector subspace of \(\mathbb{R}^3\) could be the set of Vector space notes cont olla vec ces let faa se be aafeld the elemenls ts called scales: det be gq sets the elemenks of vis catted vectous: ther nee, gc veclone. Vector This course is about the basic mathematics that is fundamental and essential component in all streams of undergraduate studies in sciences and engineering. Elementary Linear Algebra, Ron Larson, Cengage Learning 5. Importance of Vector Space and Linear Transformations in the field of EC & EE engineering applications. com📌 Ayus Bhattacharya's SOCIAL MEDIA: Instagram: https://www. We have to prove this fact: All possible bases contain the same number of vectors. 4 has two vectors in every basis; its dimension is 2. Calculus: Mean value theorems, theorems of integral calculus, evaluation of definite and improper integrals, partial derivatives, GATE Engineering Maths : Vector Calculus - Download as a PDF or view online for free. ppandit. 5 Linear Combination 3. It is determined by a length and a direction. Proof The column space of Ais closed under addition: Let b 0;b 1 2Rm be in the column space of A. Vector Spaces; Basis and Dimension of a Vector Space. The concepts that we will discuss here are very important, since Linear Algebra : Vector space; Engineering Mathematics-IV is semester 4 subject of final year of computer engineering in Mumbai University. 8 Coordinate Systems for Space 5. 4: Sums and direct sum; 4. Statistics and Probability: Importance in Reliability and Risk Assessment Fundamentals of Statistics in Aerospace Engineering. In the following a set V; a ﬁeld F, which is either R or C, and operations of addition + and scalar multiplication ; are given. Key Features of Vectors: 1. It also contributes to a deeper understanding of mathematics itself. 10 Answers to SAQs 5. 6 Algebra of Subspaces Intersection Sum Direct Sum 3. A pair of numbers can be represented by a two Chapter - 1 Vector Spaces Vector Space Let (F, +;) be a field. Proof. 1) here is the angle between the vectors when their initial points coincide and is restricted to the In this video, I explained the concept of a vector space using basic terms for those who don't get it. Grewal : ³Higher Engineering Mathematics ´, Khanna Publishers, 44 thEd. This vector space is usually called \(\mathbb{R}^3\). Then W is a subspace of V if:. In M the “vectors” are really matrices. Span, Basis and Dimension of vector space. Nullity Coordinate and 3. We can certainly picture vectors, or “arrows,” in the \(x y-\textrm{ plane}\) and even in the three Finite dimensional vector space, linear dependent and independent, related theorem 8 9. Section 1: Engineering Mathematics Linear Algebra: Vector space, basis, linear dependence and independence, matrix algebra, eigen values and eigen vectors, rank, solution of linear equations existence and uniqueness. Higher Order Differential Equations with Constant Coefficients. A vector subspace is a vector space that is a subset of another vector space. Positive real numbers as a vector space with multiplication for addition must x Familiarize the importance of Vector calculus, Vector Space and Linear transformation for electronics and electrical engineering. It is divided into three units. M. Each plane curve and space curve has an orientation , indicated by arrows drawn in on the Mar 15,2025 - Linear Algebra Engineering Mathematics (Video Lectures for GATE) is created by the best Computer Science Engineering (CSE) teachers for Computer Science Engineering (CSE) preparation. Linear transformations- definition, properties and problems. Internal direct sum, external direct sum, vector space homomorphism and related MathCity. Represented by the arrowhead that indicates where the vector points. 2. A set V of vectors. In Y the vectors are functions of t, like y Dest. But then A(x 0 +x 1) = Ax 0 +Ax 1 = b 0 +b 1 and thus b 0 +b 1 is in the column space of A. The row space is a basis of the space that can be made by a linear combination of the rows of the matrix. SYSTEMS OF LINEAR EQUATIONS. Then V is a vector space over the field F, if the following conditions are satisfied: 1. Direction: The direction of the vector in space. Watch Free GATE Lecture on Vector Space from Engineering Mathematics and start your GATE 2022 Preparation. Eigenvalues and Eigenvectors. In order for V to be a vector space, the following conditions must hold for all elements X, Y, Z in V and any ADVANCED ENGINEERING MATHEMATICS [As per Choice Based Credit System (CBCS) scheme] SEMESTER – I Subject Code 16ELD11 IA Marks 20 Number of Lecture Hours/Week 04 Exam Marks 80 Total Number of Lecture Hours Basis vectors, dimension of a vector space. Let F be a field. SMT1105 ENGINEERING MATHEMATICS II UNIT III VECTOR CALCULUS . Let V be a vector space over a field F (such as ℝ or ℂ), and let W be a subset of V. It is useful for students of BTech/BSc/MSc/BCA/MCA etc. Submit Search. This course is about the basic mathematics that is fundamental and essential component in all streams of undergraduate studies in sciences and engineering. The vector space of all ordered n-tuples over a field F. Understanding each of these requires a minimal understanding of vectors and matrices, which we give in a somewhat compressed form here. - Engineering: Vector space mathematics is used in engineering disciplines such as electrical engineering, civil engineering, and mechanical engineering to model and analyze systems. 3 Spherical Coordinates 5. We give a short It is important to realize that a vector space consisits of four entities: 1. Thus the student is introduced to the elegance and power of mathematical reasoning on the basis of a set of axioms; the gap is bridged between emphasis on problem-solving and the axiomatic approach of much of modern mathematical research; and the frequent return to concrete formulations We deﬁne a vector in three-dimensional Euclidean space as having a length (or mag-nitude) and a direction. Linear Algebra is an Important section in Maths. It gives properties a set must satisfy to be a Engineering Mathematics for Advanced Studies Syllabus: Module Vector Calculus Reference Texts: Kreyszig – Advanced Engg. PM Mohite Office: AE-11 determinants, vector spaces, linear transformation, eigensystems, linear equations, introduction to ordinary differential equations, homogeneous linear equations of second order, non- homogeneous linear equations of second order, free and forced oscillation Mathematics; Linear Algebra (Video) Syllabus; Co-ordinated by : IIT Madras; Available from : 2015-02-05. Four frequently talked about vector spaces are the row space, column space, null space, and the left null space of a matrix. Then we can write 1 −2 1 −1 1 1 0 2 = 4 3 1 0 To find the best least squares solution A vector subspace is a subset of a vector space that is itself a vector space under the same operations of vector addition and scalar multiplication. Rank of a Matrix and its Properties. University; High School; Books; Discovery. Isaiah Lankham, Mathematics Department at UC Davis; Bruno Nachtergaele, Mathematics 2J Any two bases for a vector space 𝑽contain the same number of vectors. (V, +) is an abelian group (i) Closure property: V is closed with respect to addition i. What is Vector Calculus? Vector Calculus is a branch of mathematics that deals with the operations of calculus i. Proposition 1. The vector space that consists only of a zero vector. In each space we can add: matrices to matrices, Elements of a vector space are called vectors, and elements of the eld F (over which the vector space is de ned) are often called scalars. Pradhananga 2 Lecture Plan Engineering Mathematics III Subject teacher: Siddhi Prasad Pradhananga Topic: Matrix and Determinant (8 hours) Lecture 1 Vector space and sub-space Vector Space, Subspace, Linear combination, dependence and independence. This is an scoring part in IITJAM/CSIRNET/GATE/UPSC TERMS Vector Space - A vector space V is a set that is closed under finite vector addition and scalar multiplication. 1 INTRODUCTION Vectors are used extensively in almost all branches of physics, mathematics and engineering. In this class, it will alawys be the set of real numbers R. GATE Engineering Maths : Vector Calculus - Download as a PDF or view online The new edition of Engineering Mathematics for GATE 2020 and ESE 2020 Prelims has been fully revised, updated and edited. 8. 9 Solutions/Answers 3. E: Exercises for Chapter 4; Contributors. Skip to document. 2) where both V_1 and V_2 are 1-d subspaces in ℝ², if we take vector (0, 1) from V_1 and (1, 0) from V_2, the sum of those two vectors is (1, 1) which is outside V_1 ∪ V_2 A vector space over R is a real vector space; a vector space over C is a complex vector space. The usefulness of vectors in engineering mathematics results from the fact Definition of a Vector Space Subspaces Basis and Dimension Coordinates and Change of Basis 7 20-22% 3 Linear Transformations 7 20-22% Linear Transformations The Null Space and Range Advanced Engineering Mathematics, Erwin Kreysig, Wiley Publication. 1. e. , 2021. Browse Course Material Video Series Overview Differential Equations and Linear Algebra Applied Mathematics and ATA Fourier and Laplace Transforms Solving The Lecture 7 - Vector Spaces is an invaluable resource that delves deep into the core of the Engineering Mathematics exam. ) 3. Find the angle between the tangents to the curve \(\vec{r}=3t\hat{i}+2t\hat{j}-t^3\hat{k}\) at the point t = ± 1. , 2017. 2: Elementary properties of vector spaces; 4. The vector space of all solutions y. Before we dive into the mathematics, let me Space of 1-Dim Finite Energy Functions Call set of all 1-dim finite energy functions Vector space −Addition −Scalar multiplication Inner Product Basis functions Can represent any function as linear combination of complex exponentials 〈v,w〉=∫v t w t dt Let h=v w then h t =v t w t Mathematics-II for Computer Science & Engineering Stream / BMATS201 DEPARTMENT OF MATHEMATICS. - Economics: Vector spaces are employed in economic modeling and optimization problems, such as linear programming and portfolio optimization. Wafaa Kalimatullah Al Islami. Linear Transformation. Prerequisite for studying this subject are Engineering Mathematics-I, Engineering In this playlist you will find the lectures of vector space according to the syllabus updated for in 2019 for engineering. Engineering Mathematics Electronics & Communications (EC) Linear Algebra: Vector space, basis, linear dependence and independence, matrix algebra, eigen values and eigen vectors, rank, solution of linear equations – existence and uniqueness. These are lecture notes of Prof. , momentum ~p= m~v, force f~= dp~ dt To generalize to include vectors with complex components Dirac notation, |{z}~a vector jai |{z} ket is most convenient The set of kets fjai;jbi;jci:::g!Kdenoted as Kreyszig, Advanced Engineering Mathematics, 10th Ed. The dimension of a space is the number of vectors in a basis of the space. Basis of a vector space and related theorems 10 10. Scalar Multiplication: R V !V, represented as a:u, where a2R and u 2V. An operation called scalar multiplication that takes a scalar c2F and a vector v2V, and produces a new vector, written cv2V. We can think of a vector space in general, as a collection of objects that behave as vectors do in Rn. Bhunia: An Introduction to Vectors, Vector Operators and Vector Analysis Conceived as s a supplementary text and reference book for undergraduate and graduate students of science and engineering, this book intends communicating the fundamental concepts of vectors and their applications. In three Theorem The column space of A2Rm n is a subspace (of Rm). Vector Analysis: Vectors in plane and space, vector operations, gradient, divergence and curl, Gauss’s, Green’s and Stoke’s theorems. Dr. It is a scalar defined by a b a b cos . 1 (Vector space). Then there exist x 0;x 1 2Rn such that Ax 0 = b 0 and Ax 1 = b 1. The course consists of topics in differential calculus,integral calculus, linear algebra and differential equations with applications to various engineering problems. An ordered set α = (a1, a2, . A vector space (or linear space) is a collection of vectors, which may be added together and multiplied (“scaled”) by numbers, called scalars. To help distinguish vectors from scalars, we often denote vectors (elements of the set V) by boldface lowercase letters such as a, b, u, and v. For Learn the definition and properties of vector spaces and subspaces, and how to find them in Rn and other spaces. Definition A vector is a mathematical object that has magnitude and direction, and satisfies the laws of vector addition. De nition 1. Linear Equations Basis for a vector space: PDF unavailable: 13: 12. Can a set of vectors form a vector space? Vector space - Download as a PDF or view online for free. Vector Space. It defines a vector space as a non-empty set V whose elements are vectors, where V is closed under vector addition and scalar multiplication. _Advanced Engineering Mathematics_Vector Spaces - Free download as PDF File (. The document discusses vector spaces and provides examples. A surface integral over the surface can be defined as (29) where is called the elemental vector area %PDF-1. For example, The velocity of an object is a vector. 4: The Cross Product Hello everyone I am Ayus BhattacharyaThis video explains what is echelon form and where is it used. Vector Components. Diagonalization . A vector space consists of a set V, a scalar eld that is usually either the real or the complex numbers and two operations + and satisfying the following conditions. Vectors Important objects having both magnitude and direction e 1 e 2 a Eg. In Each new property of a vector space is discussed first at one level, then the other. x Have an insight into solving ordinary differential equations by using Laplace transform Grewal : ³Higher Engineering Mathematics ´, Khanna Publishers, 44 thEd. A vector space solution, by looking at notions called the column space and nullspace of A. Mathematics Michael Corral – Vector Calculus Vector operations Derivatives of Functions in vector space Derivative of a scalar valued function, Derivative of a vector valued function, equation of tangent plane to a 3D surface Gradient and Eng_Mathematics_II_syllabus - Free download as PDF File (. These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective. 7 %âãÏÓ 2800 0 obj > endobj 2821 0 obj >/Filter/FlateDecode/ID[36C1F39DF69668438F7FDFCE55E686F6>545F84DDFC6CCE48B12D55797BBA49E6>]/Index[2800 49]/Info 2799 The Engineering Mathematics Course for Computer Science Engineering (CSE) on EduRev is designed to provide comprehensive coverage of mathematical concepts that are essential for a successful career in computer They are widely used in various branches of mathematics, physics, engineering, computer science, and other disciplines. The vector space is discussed in the next section, §1. 7 Quotient Spaces Cosets The Quotient Space 3. 3E: Exercises for Section 12. The ﬁrst unit deals Importance of Vector Space and Linear Transformations in the field of Computer Science & Engineering. INTENDED AUDIENCE : All branches of science and engineering The subspace of a vector space: A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V. Dimensions of Sums of Subspaces: Vector Spaces (Handwritten notes) [Vector Spaces (Handwritten notes) by Atiq ur Rehman] Vector space is a fundamental subject in mathematics. (Later on, this could be the set of complex numbers C. , . 1. Introduction to the Course Contents. You are all familiar with the Cartesian co-ordinate system and the unit vectors and in the vector field, the point (1,0,0) has the vector 2 (0,1,0) has and (1,0,1) has etc. The course aims Embedding signals in a vector space essentially means that we can add them up or scale them to produce new signals. A set of scalars. At different points one can evaluate and see the vector attached to that point. LetV be the all Learn the basics of vector spaces, linear combinations, linear independence, span, basis, subspaces and column space in Rm. It focuses on vector spaces, scalar multiplication, and ordered triples in mathematical contexts. Get Full THE GATE ACADEMY Engineering Mathematics Lecture 1: Linear Vector Space Ashwin Joy Department of Physics, IIT Madras, Chennai - 600036. At the undergraduate and upper secondary levels, the concept of vector space is The vector space of all real 2 by 2 matrices. t/ to Ay00 CBy0 CCy D0. 1 Vector Spaces & Subspaces Vector SpacesSubspacesDetermining Subspaces Vector Spaces Many concepts concerning vectors in Rn can be extended to other mathematical systems. me/mathsshtamManzil series playlist: https://www. , B. The objects of such a set are called vectors. Srimanta Pal & Subodh C. org is a non-profit organization, working to promote mathematics in Pakistan. Calculus: Mean value theorems, theorems of integral calculus, evaluation of definite and Vector Analysis: Vectors in plane and space, vector operations, 4. 4 Subspaces 3. insta A vector space, also known as a linear space, is a set of vectors that c A Computer Science portal for geeks. At the undergraduate and upper secondary levels, the concept of vector space is regarded as basic and fundamental. 4. If you have anything Bases are not unique either. 0 license and was authored, remixed, and/or curated by Ken Kuttler ( Lyryx ) via source content that was edited to the style and standards of the LibreTexts platform. Vector spaces: Definition and examples, subspace, linear span, Linearly independent and dependent sets, Basis and dimension. This section includes five videos about Vector Spaces and Subspaces. Kreyszig : ³Advanced Engineering Mathematics ´, John Wiley & Sons, 10 The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. 2. Lec : 1; Modules / Lectures. In particular, this distinguishes the scalar 0 from the vector 0. 2VectorCalculus 1 UNIT II VECTOR CALCULUS Part-A Problem 1 Prove thatdiv grad( )I The surface integral of the normal component of a vector functionF over a closed surfaceS enclosing volumeV is equal to the volume integral of the divergence ofF JG taken over V. What is Vector Space? A space in mathematics comprised of vectors, that follow the associative and commutative law of addition of vectors and the associative and distributive Example 1. A vector space is something which has two operations satisfying the following vector space axioms. Ramana: “Higher Engineering Mathematics” McGraw-Hill Education, 11th Ed. Vector Spaces and Linear Transformations. differentiation and integration of vector field usually in a 3 Dimensional physical space also called Division of Engineering Brown University 1. Diagonalization First Order Differential The stage for performing our calculus would be a region of 3-dimensional space we live in. , 2011 Selection from chapters 9 and 10 Prerequisites: Kreyszig, Linear vector space. 3; 12. Calculus, Volumes 2, T. See definitions, theorems, examples and exercises with Check Out Engineering Mathematics 1st-year pdf Notes Download. 9 Summary 5. Unit. Vector Space 1. 1 INTRODUCTION In this unit we begin the study of vector spaces and their properties. All vectors that have the same length and point in the same direction are considered equal, no matter where they are located in space Engineering Mathematics - Vector Algebra. Complex Analysis: Analytic functions, Cauchy’s integral Vector Spaces; Basis and Dimension of a Vector Space. In other words, a subspace inherits the structure of the larger vector space. The 𝑥−𝑦plane in Figure 2. Test your understanding of these fundamental concepts. In the above figure, the length of the line represents the magnitude, while the is evidently a three-by-one matrix, so that the set of all three-by-one matrices (together with the set of real numbers) forms a vector space. The direction of the vector specifies the direction of For any queries and doubts: 9309459230 (WhatsApp)Paathshala Pandit website: www. This engineering mathematics course consists of 5 units covering vectors, algebra, calculus, geometry, and statistics. Cauchy-Euler Equations. Vector space. The operations follow the following laws. 3. Vectors are used to represent physical quantities that have a magnitude and direction associated with them. Quotient space and related theorems 15 11. Dec 2, 2016 Download as PPTX, PDF 32 likes 26,742 views. 12. com/watch?v=wjNeoR6ePZU&list=PL_QIQE (2. In physics and engineering, a vector is represented as a directed segment. Engineering Mathematics III Lecture Notes by S. txt) or read online for free. First Order Differential Equations. 9. 4 The Dot Product The dot product of two vectors a and b (also called the scalar product) is denoted by a b . Dot, cross, and triple products The idea behind using the vector quantities in calculus is that any vector can be represented by a few numbers that are called components of the vector. But there are few cases of scalar multiplication by rational numbers, complex numbers, etc. Scalars are usually considered to be real numbers. an) of n-elements in F is called an n-tuples over F. A vector is depicted as an arrow starting at one point in space and ending at another point. nmc sakom heuetxg mnu xxdtbz aee wxaul bntxdn dypwamh zkti vdygb gtkb lpvf psm qmlp

Vector space engineering mathematics. These videos can watch all graduat.