Vector space engineering mathematics. This is an scoring part in IITJAM/CSIRNET/GATE/UPSC .

Vector space engineering mathematics 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. For example, The velocity of an object is a vector. 2: Elementary properties of vector spaces; 4. t/ to Ay00 CBy0 CCy D0. The 𝑥−𝑦plane in Figure 2. Importance of Vector Space and Linear Transformations in the field of EC & EE engineering applications. 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. The ﬁrst unit deals Importance of Vector Space and Linear Transformations in the field of Computer Science & Engineering. Direction: The direction of the vector in space. 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. 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. This course is about the basic mathematics that is fundamental and essential component in all streams of undergraduate studies in sciences and engineering. 3: Subspaces; 4. 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. Let V be a non empty set whose elements are vectors. 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. 4: Sums and direct sum; 4. sets are subsets of vector spaces or linear mapping is function from a vector space into vector space. 6 Algebra of Subspaces Intersection Sum Direct Sum 3. 5 Linear Combination 3. 1: Deﬁnition of vector spaces; 4. The course aims Embedding signals in a vector space essentially means that we can add them up or scale them to produce new signals. Assignment-Problems: Vector spaces. 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. e. Then V is a vector space over the field F, if the following conditions are satisfied: 1. 2. Vector Space 1. We can think of a vector space in general, as a collection of objects that behave as vectors do in Rn. Vector space. Unit. At different points one can evaluate and see the vector attached to that point. Vector Spaces and Linear Transformations. P. 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. 3E: Exercises for Section 12. , . Ramana: “Higher Engineering Mathematics” McGraw-Hill Education, 11th Ed. 8 Summary 3. It is useful for students of BTech/BSc/MSc/BCA/MCA etc. - Engineering: Vector space mathematics is used in engineering disciplines such as electrical engineering, civil engineering, and mechanical engineering to model and analyze systems. But there are few cases of scalar multiplication by rational numbers, complex numbers, etc. 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. Then W is a subspace of V if:. In this course you will be expected to learn several things about vector Vector Spaces; Basis and Dimension of a Vector Space. Quotient space and related theorems 15 11. 4 has two vectors in every basis; its dimension is 2. , 2021. 12. Then there exist x 0;x 1 2Rn such that Ax 0 = b 0 and Ax 1 = b 1. In physics and engineering, a vector is represented as a directed segment. 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. 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. 4 Subspaces 3. This vector space is usually called \(\mathbb{R}^3\). 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 vector space over a field F (such as ℝ or ℂ), and let W be a subset of V. A vector subspace is a vector space that is a subset of another vector space. E. Sc. The document discusses vector spaces and provides examples. This Playlist of Vector Space is helpful for students preparing for B. The direction of the vector specifies the direction of For any queries and doubts: 9309459230 (WhatsApp)Paathshala Pandit website: www. 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. 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. We have provided Mathematics 1st Year Study Materials and 1. Linear transformations- definition, properties and problems. Elementary Linear Algebra, Ron Larson, Cengage Learning 5. An operation called scalar multiplication that takes a scalar c2F and a vector v2V, and produces a new vector, written cv2V. A vector is depicted as an arrow starting at one point in space and ending at another point. x Have an insight into solving ordinary differential equations by using Laplace transform Grewal : ³Higher Engineering Mathematics ´, Khanna Publishers, 44 thEd. The whole book has been divided into topicwise sections. 1. 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. Get Full THE GATE ACADEMY Engineering Mathematics Lecture 1: Linear Vector Space Ashwin Joy Department of Physics, IIT Madras, Chennai - 600036. 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. 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. 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. Represented by the arrowhead that indicates where the vector points. an) of n-elements in F is called an n-tuples over F. Definition A vector is a mathematical object that has magnitude and direction, and satisfies the laws of vector addition. Lec : 1; Modules / Lectures. Calculus, Volumes 2, T. 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. 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. , 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. Key Features of Vectors: 1. 1 Vector Spaces & Subspaces Vector SpacesSubspacesDetermining Subspaces Vector Spaces Many concepts concerning vectors in Rn can be extended to other mathematical systems. 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. In this class, it will alawys be the set of real numbers R. Statistics Solved Example of Application of Vector Calculus in Engineering Mathematics. Dimension of a vector space: PDF unavailable: 14: 13. 3 Further Properties of a Vector Space 3. De nition 1. The dimension of a space is the number of vectors in a basis of the space. E: Exercises This page titled 9: Vector Spaces is shared under a CC BY 4. 10 Answers to SAQs 5. Proof The column space of Ais closed under addition: Let b 0;b 1 2Rm be in the column space of A. , B. 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. LetV be the all Learn the basics of vector spaces, linear combinations, linear independence, span, basis, subspaces and column space in Rm. 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. Submit Search. i. 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. We give a short It is important to realize that a vector space consisits of four entities: 1. , 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. 3 Spherical Coordinates 5. ppandit. , M. 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. youtube. A set V of vectors. Vector This course is about the basic mathematics that is fundamental and essential component in all streams of undergraduate studies in sciences and engineering. If you have anything Bases are not unique either. Watch Free GATE Lecture on Vector Space from Engineering Mathematics and start your GATE 2022 Preparation. Proposition 1. , 2017. 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. (V, +) is an abelian group (i) Closure property: V is closed with respect to addition i. Vector Components. Vectors Important objects having both magnitude and direction e 1 e 2 a Eg. These are lecture notes of Prof. 2 Vectors and Matrices It is easiest to think of a vector as a generalisation of a single number. _Advanced Engineering Mathematics_Vector Spaces - Free download as PDF File (. differentiation and integration of vector field usually in a 3 Dimensional physical space also called Division of Engineering Brown University 1. - Economics: Vector spaces are employed in economic modeling and optimization problems, such as linear programming and portfolio optimization. me/mathsshtamManzil series playlist: https://www. 1 Cartesian Coordinates 5. 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. Rank of a Matrix and its Properties. 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. 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. A vector space is something which has two operations satisfying the following vector space axioms. Complex Analysis: Analytic functions, Cauchy’s integral Vector Spaces; Basis and Dimension of a Vector Space. It is divided into three units. Skip to document. 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. For any pair of elements ~x;~y2Vthe vector sum ~x+ ~ybelongs to V. Can a set of vectors form a vector space? Vector space - Download as a PDF or view online for free. 9 Summary 5. Scalar Multiplication: R V !V, represented as a:u, where a2R and u 2V. Understanding each of these requires a minimal understanding of vectors and matrices, which we give in a somewhat compressed form here. A vector addition MATHEMATICS FOR AEROSPACE ENGINEERS L-T-P-D: 3-0-0-0 Units: 4 Course Instructor: Dr. 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. It is a scalar defined by a b a b cos . 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. In M the “vectors” are really matrices. 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. We have to prove this fact: All possible bases contain the same number of vectors. 9. 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. Vector Space. Diagonalization First Order Differential The stage for performing our calculus would be a region of 3-dimensional space we live in. Span, Basis and Dimension of vector space. The vector space of all ordered n-tuples over a field F. In Each new property of a vector space is discussed first at one level, then the other. The vector space that consists only of a zero vector. 8. In the following a set V; a ﬁeld F, which is either R or C, and operations of addition + and scalar multiplication ; are given. Higher Order Differential Equations with Constant Coefficients. . Grewal : ³Higher Engineering Mathematics ´, Khanna Publishers, 44 thEd. Vectors are used to represent physical quantities that have a magnitude and direction associated with them. M. (Later on, this could be the set of complex numbers C. Dec 2, 2016 Download as PPTX, PDF 32 likes 26,742 views. It gives properties a set must satisfy to be a Engineering Mathematics for Advanced Studies Syllabus: Module Vector Calculus Reference Texts: Kreyszig – Advanced Engg. 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. Scalars are often real numbers, but can also be complex numbers. Statistics and Probability: Importance in Reliability and Risk Assessment Fundamentals of Statistics in Aerospace Engineering. In Y the vectors are functions of t, like y Dest. 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. 9 Solutions/Answers 3. Test your understanding of these fundamental concepts. Tech (Engineering) & other competitive exams like IIT-JAM, CSIR NET, GATE. In particular, this distinguishes the scalar 0 from the vector 0. Prove that the Subset 𝑊 = {(𝑥, 𝑦, 𝑧) \𝑥 − 3𝑦 + 4𝑧 = 0} of the vector space R 3 is a subspace of R 3. A surface integral over the surface can be defined as (29) where is called the elemental vector area %PDF-1. 1. Nullity Coordinate and 3. The vector space is discussed in the next section, §1. It is determined by a length and a direction. Wafaa Kalimatullah Al Islami. 1 INTRODUCTION In this unit we begin the study of vector spaces and their properties. 2. Isaiah Lankham, Mathematics Department at UC Davis; Bruno Nachtergaele, Mathematics 2J Any two bases for a vector space 𝑽contain the same number of vectors. Calculus: Mean value theorems, theorems of integral calculus, evaluation of definite and Vector Analysis: Vectors in plane and space, vector operations, 4. MODULE – III. 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. pdf), Text File (. These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective. com📌 Ayus Bhattacharya's SOCIAL MEDIA: Instagram: https://www. Linear Algebra is an Important section in Maths. Cauchy-Euler Equations. 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. 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. Linear Equations Basis for a vector space: PDF unavailable: 13: 12. 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. Scalars are usually considered to be real numbers. The row space is a basis of the space that can be made by a linear combination of the rows of the matrix. For Learn the definition and properties of vector spaces and subspaces, and how to find them in Rn and other spaces. In three Theorem The column space of A2Rm n is a subspace (of Rm). 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. The course consists of topics in differential calculus,integral calculus, linear algebra and differential equations with applications to various engineering problems. 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. Diagonalization . Vector spaces: Definition and examples, subspace, linear span, Linearly independent and dependent sets, Basis and dimension. 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 (. Basis of a vector space and related theorems 10 10. First Order Differential Equations. (1. University; High School; Books; Discovery. 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. 7 Quotient Spaces Cosets The Quotient Space 3. This section includes five videos about Vector Spaces and Subspaces. 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. 3. 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. It also contributes to a deeper understanding of mathematics itself. 4. Linear Transformation. Vector Spaces; Basis and Dimension of a Vector Space. A set of scalars. Internal direct sum, external direct sum, vector space homomorphism and related MathCity. E: Exercises for Chapter 4; Contributors. At the undergraduate and upper secondary levels, the concept of vector space is The vector space of all real 2 by 2 matrices. 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. Proof. Eigenvalues and Eigenvectors. 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. org is a non-profit organization, working to promote mathematics in Pakistan. with vector spaces. The vector space of all solutions y. Eigenvalues and Eigen vectors. This engineering mathematics course consists of 5 units covering vectors, algebra, calculus, geometry, and statistics. A vector space solution, by looking at notions called the column space and nullspace of A. At the undergraduate and upper secondary levels, the concept of vector space is regarded as basic and fundamental. It focuses on vector spaces, scalar multiplication, and ordered triples in mathematical contexts. The operations follow the following laws. 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. What is Vector Calculus? Vector Calculus is a branch of mathematics that deals with the operations of calculus i. 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. 1 (Vector space). Engineering Mathematics III Lecture Notes by S. Let F be a 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. Vector Analysis: Vectors in plane and space, vector operations, gradient, divergence and curl, Gauss’s, Green’s and Stoke’s theorems. Srimanta Pal & Subodh C. 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. 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. ) 3. 8 Coordinate Systems for Space 5. 4 The Dot Product The dot product of two vectors a and b (also called the scalar product) is denoted by a b . These videos can watch all graduat SOLUTION: Let the equation of the best fit line be of the form 𝒃= + 𝒕. Linear 5. A pair of numbers can be represented by a two Chapter - 1 Vector Spaces Vector Space Let (F, +;) be a field. See definitions, theorems, examples and exercises with Check Out Engineering Mathematics 1st-year pdf Notes Download. txt) or read online for free. Four frequently talked about vector spaces are the row space, column space, null space, and the left null space of a matrix. 3; 12. The objects of such a set are called vectors. 4: The Cross Product Hello everyone I am Ayus BhattacharyaThis video explains what is echelon form and where is it used. 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. Kreyszig : ³Advanced Engineering This concept is explored in this section, where the linear transformation now maps from one arbitrary vector space to another. insta A vector space, also known as a linear space, is a set of vectors that c A Computer Science portal for geeks. , 2011 Selection from chapters 9 and 10 Prerequisites: Kreyszig, Linear vector space. In other words, a subspace inherits the structure of the larger vector space. 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. 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. 1 INTRODUCTION Vectors are used extensively in almost all branches of physics, mathematics and engineering. com/watch?v=wjNeoR6ePZU&list=PL_QIQE (2. SMT1105 ENGINEERING MATHEMATICS II UNIT III VECTOR CALCULUS . Dr. It allows us to perform all operation on Vector Space Theory A course for second year students by Robert Howlett typesetting by TEX. 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. SYSTEMS OF LINEAR EQUATIONS. 2 Cylindrical Coordinates 5. This number, which is shared by all bases and expresses the number of “degrees of freedom” of the space, is the dimensionof 𝑽. In Z the only addition is 0 C0 D0. Introduction to the Course Contents. 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. adh wingmu yqbxt cazh iexru qzlsg wzrrr hqvyep nmhrjw okfhf qtxkdzu ycmmr qadytj kwvoqg sje