#### review-1

UCLA, MATH 33a
Excerpt: ... Math 33A Spring 2008 First mid-term exam study guide The first mid-term exam will cover the material from chapters 13 in the textbook (this includes Section 3.4, which was covered in lecture on Monday, April 21). 1 Essential notions and techniques The following is a list of essential notions you should know, and techniques you should be able to use, from the first three chapters of the textbook. Be able to solve a system of linear equations using Gauss-Jordan elimination; Be able to transform a matrix into reduced row-echelon form; Be able to multiply a matrix with a vector; Be able to find the matrix of a linear transformation , given the values T (e1 ), T (e2 ), . . ., T (en ), or from the values T (v1 ), T (v2 ), . . ., T (vn ) for some vectors vi simply-related to the vectors ei ; Be able to determine the rank of an n m matrix; Know how the rank of an n m coefficient matrix relates to the number of solutions that the corresponding system has; Know that a function T from Rm to ...

#### rev2

Kent State, MATH 41021
Excerpt: ... Math 41021 Exam II Review Exam II will be given in class on Wednesday, April 1, 2009. It will cover Chapters 5 and 6 and 7.17.7, Homework #5#8, and material from class February 18 through March 16 (pages 3466 of the lecture notes). The exam will consist of statements of definitions and theorems (from the list below), computational problems, and proofs similar to those in the homework. The following is an outline of topics and types of problems that may be on the exam. Definitions to State: Inner Product Orthogonal Complement For a Linear Transformation : Norm of a Vector Projection of v along u Kernel Angle Between Vectors Projection of v onto W Image Orthogonal Vectors Linear Transformation Rank Orthogonal Set Isomorphism Nullity Orthonormal Set Similar Matrices Rank + Nullity Theorem Singular/Nonsingular Inner Product Spaces: Definition of inner product, dot product on Rn , other examples, definition and properties of norm, Cauchy-Schwarz Inequality, computations of inner products, lengths ...

#### Math221Lecture007BSlides

UMBC, MATH 221
Excerpt: ... Transformations Linear Transformation s Lecture 7: Linear Transformation s Transformations Mapping Rn to Rm Matrix Transformations Geometry of Matrix Transformations Linear Transformation s Definition of a Linear Transformation Two Basic Results on Linear Transformation s The Superposition Principle Clint Lee Math 221 Lecture 7: Linear Transformation s 1/9 Transformations Mapping Rn to Rm Linear Transformation s Transformations from Rn to Rm If A is an m n matrix, the vector-matrix product Ax transforms or maps a vector x in Rn into a vector in Rm . Clint Lee Math 221 Lecture 7: Linear Transformation s 2/9 Transformations Mapping Rn to Rm Linear Transformation s Transformations from Rn to Rm If A is an m n matrix, the vector-matrix product Ax transforms or maps a vector x in Rn into a vector in Rm . Thus x Ax is a function from Rn to a set in Rm Clint Lee Math 221 Lecture 7: Linear Transformation s 2/9 Transformations Mapping Rn to Rm Linear Transformation s Transformations from Rn to Rm I ...

#### week4

Rutgers, MATH 350
Excerpt: ... 640:350: LINEAR ALGEBRA HIGHLIGHTS OF LECTURES FOR WEEK 4 Sections to be covered: 2.1, 2.2, 2.3. (1) Note that if A is an m n matrix, then T : Rn Rm defined by T (x) = Ax for any vector x Rn is a linear transformation , yet for any non-zero vector b, x Ax + b does not define a linear transformation . b x (2) Note that both T1 (f ) = a f (x) dx and T2 (f ) = a f (x ) dx define linear transformation s, but T1 can be regarded as a linear transformation from C[a, b] to R, but T2 has a different codomain: it can be regarded as a linear transformation from C[a, b] to C[a, b]. The precise range of T2 is in fact a subspace of C[a, b]: T2 (C[a, b]) = {g C 1 [a, b] : g(a) = 0}. (3) Theorem 2.2 provides a method for finding a spanning set for the range R(T ) of a linear transformation T . Then the dimension and a basis of R(T ) can be found from the method of Theorem 1.9. (4) The dimension of R(T ) is related to the dimension of the domain and the dimension of the null space N (T ) through the dimension theorem, Theo ...

#### Math221Lecture008BSlides

UMBC, MATH 221
Excerpt: ... The Matrix of a Linear Transformation Existence and Uniqueness Questions Lecture 8: The Matrix of a Linear Transformation The Matrix of a Linear Transformation The Identity Matrix and Its Columns The Matrix of a Linear Transformation , Theorem 1.10 Geometric Linear Transformation s Existence and Uniqueness Questions Onto Transformations One-to-One Transformations Testing for Linear Transformation s, Theorems 1.11 & 1.12 Clint Lee Math 221 Lecture 8: The Matrix of a Linear Transformation 1/19 The Matrix of a Linear Transformation The Identity Matrix and Its Columns Existence and Uniqueness Questions The Identity Matrix The Identity Matrix Clint Lee Math 221 Lecture 8: The Matrix of a Linear Transformation 2/19 The Matrix of a Linear Transformation The Identity Matrix and Its Columns Existence and Uniqueness Questions The Identity Matrix The Identity Matrix The n n matrix that has ones along the main diagonal and zeros elsewhere is the identity matrix, denoted In . Clint Lee Math 221 Lecture ...

#### lab1

Grand Valley State, MATHFEST 06
Excerpt: ... function f (x) = mx that you might encounter in Math 201; our aim today is to consider it carefully. Since we have identified complex numbers with two dimensional real vectors, we may also think of Tm as being a function Tm : R 2 R 2 . Viewed in this way, the function is, in fact, a linear transformation (you may verify it if you like). Remember that any such linear transformation is defined by a 2 2 matrix Mm . What is the matrix Mm ? (If you don't remember how to find the matrix given by a linear transformation , ask me.) What is det(Mm ) and what is the geometric significance of this determinant in terms of m? 1 Note for Math 310 students: The association m Mm defines an isomorphism between the field of complex numbers and a sub-field of the ring of 2 2 matrices. This gives a way of describing the complex numbers that answers the sticky question "What is i?" 2. Now go to the web page: http:/merganser.math.gvsu.edu/m402/lab1.html Here you will find a diagram, labelled "Part I," that allows you to ...

#### lec03

Berkeley, MATH 110
Excerpt: ... MATH 110 Lecture Notes 3 GSI Carter June 25, 2008 1 Linear Transformation s Let V and W be vector spaces over a field F . Then a function T : V W is called a linear transformation if for every x, y V and every F , we have T (x + y) = T (x) + T (y) T (x) = T (x) Such a function can also be referred to as an F - linear transformation to indicate which field F is being used. Some examples: Multiplication by a matrix A. Differentiation. Integration on a fixed interval [a, b]. Theorem. Let T : V W be linear. Then the null space N (T ) is a subspace of V , and the range R(T ) is a subspace of W . 1 Theorem. Let T : V W be linear. Then T maps any generating set for V to a generating set for R(T ). Theorem. Let T : V W be linear. Then dim N (T ) + dim R(T ) = dim V. We define rank(T ) to be dim R(T ) and nullity(T ) to be dim N (T ). Theorem. Let T : V W be linear. Then T is one-to-one if and only if N (T ) = {0}. Theorem. A linear transformation is uniquely determined by its behavior on a ...

#### L25_KernelAndRange

Laurentian, MATH 1057
Excerpt: ... Kernel and Range Lecture 25, MATH 1057E Julien Dompierre Dpartement de mathmatiques et dinformatique e e Universit Laurentienne e 19 mars 2007, Sudbury Julien Dompierre 1 Transformations (p. 100 ) Denition Let U and V be two vector spaces. A transformation T of U into V , written T : U V , is a rule that assigns to each vector u in U a unique vector v in V . The vector space U is called the domain of the transformation T and the vector space V is the codomain. We write v = T (u). v is called the image of u under the transformation T and u is called the preimage of v. The set of all images is called the range of T . The range may be the whole of V or only a part of it. The term mapping is also used for a transformation. Julien Dompierre 2 Linear Transformation (p. 314) Denition Let U and V be two vector spaces. A transformation T : U V is said to be a linear transformation if 1) T (u + v) = T (u) + T (v) for all vectors u and v in U. 2) T (cu) = cT (u) for all vector u in U ...

#### 09S-60-Lecture06

Cal Poly Pomona, SG 064747
Excerpt: ... LECTURE 6: POWERS, ELEMENTARY MATRICES MATH 60 SPRING 2009 1. Linear Transformation s Recall that an m n matrix A implicitly provides us with a function TA : Rn Rm . Indeed, the formula TA (x) = Ax provides us with a rule for transforming vectors in x Rn into certain vectors in Ax Rm . At this point, we would like to be a bit more formal in our notation and terminology. A function T : Rn Rm is usually referred to as a mapping of a transformation. Although there are many functions that one might study, linear algebra is concerned with a particularly important class of functions: Denition. A function T : Rn Rm is called a linear transformation if T (u + v) = T (u) + T (v) T (cu) = cT (u) (1) for any c R and u, v Rn . A linear transformation T : Rn Rn is called a linear operator on Rn . In particular, note that if A is an m n matrix, then the associated function TA : Rn Rm is a linear transformation . Indeed, note that the rules for matrix arithmetic imply that TA ...

#### lecture4-231-4up

Colby, MA 231
Excerpt: ... Announcements Reading Mathematics 231 Lecture 4 Liam OBrien O Today Next class M&M 1.2 M&M 1.3 M&M 1.3 45-47 4553-62 5362-71 62- 1 2 Linear Transformation s, Standardizing, and the Normal Distribution Linear transformation s: Impact on shape, center, and spread Standardizing Introduction to the normal (or Gaussian) distribution Example: Linear Transformation Temperature: Celsius to Fahrenheit 0 9 F = 32 + (0 C ) 5 Currency: Euro to U.S. dollar 1 USD = 0.814 Euros 3 4 1 Linear Transformation Lets pretend that we have nothing better to do Let but to imagine we have a set of n observations, x1,x2,xn. What we want is a set of variables, yi, related to xi by, yi = a + b(xi) Examples: Linear Transformation Temperature: Celsius to Fahrenheit 0 9 F = 32 + (0 C ) 5 y = a + bx 9 5 6 a = 32; b = 5 Examples: Linear Transformation Euros to USD Linear Transformation s yi = a + bxi A linear transformation is one that changes the data by adding a constant, multiplying by a constant, or both. ...

#### 10022008

Dallas, EE 2300
Excerpt: ... 10022008 3.4 Linear Transformation s on Rn ...

#### 09S-60-Lecture13

Cal Poly Pomona, SG 064747
Excerpt: ... LECTURE 13: SURJECTIVE LINEAR TRANSFORMATION S MATH 60 SPRING 2009 1. Linear Transformation s Recall that an m n matrix A implicitly provides us with a function TA : Rn R dened by the formala m TA (x) = Ax where x R and Ax R . A function T : Rn Rm is usually referred to as a mapping of a transformation. Although there are many functions that one might study, linear algebra is concerned with a particularly important class of functions: Denition. A function T : Rn Rm is called a linear transformation if T (u + v) = T (u) + T (v) T (cu) = cT (u) for any c R and u, v Rn . A linear transformation T : Rn Rn is called a linear operator on Rn . An important fact about linear transformation s is that they send zero vectors to zero vectors: Theorem 1. If T : Rn Rm is a linear transformation , then T (0) = 0. (1) n m A word of caution must be given regarding (1). Since T : Rn Rm , it follows that the zero vectors in (1) are not necessarily the same. Indeed, T (0) refer ...

#### sh20

Concordia Chicago, MATH 163
Excerpt: ... Sheet 20: Linear Transformation s Liz Beazley May 7, 2007 Now that we understand vector spaces, the natural thing to study is maps between vector spaces. Unless otherwise stated, always assume that V and W are arbitrary vector spaces over a eld F . Denition 1. Let V and W be vector spaces over a eld F . A map T : V W is called a linear transformation , or linear operator, if (i) T (v1 + v2 ) = T (v1 ) + T (v2 ) for all v1 , v2 V , (ii) T (v) = T (v) for all v V and F . Example 2. Consider R as a vector space over itself. Fix a Ta (x) = ax. Then Ta is a linear transformation on R. R and dene *Exercise 3. Let S denote the set of convergent sequences of real numbers. (1) Prove that S is a vector space over R. (2) Let L : S R be given by (an ) lim an . Is L a linear transformation ? n (3) Let T : S S be given by (T f )(n) = f (n + 1), where f : function. Is T a linear transformation ? N R is a Theorem 4. If T : V W is an injective linear transform ...

#### Exam2_StudyGuide

Illinois State, MATH 175
Excerpt: ... transformations. 10. Properties of linear transformation s (Theorem 2.8). 11. Relationship between matrix multiplication and linear transformation s (Theorem 2.9). 12. Finding the standard matrix for a linear transformation . 13. Denition of onto and equivalent conditions for a linear transformation to be onto (Theorem 2.10). 14. Denition of one-to-one and equivalent conditions for a linear transformation to be one-to-one (Theorem 2.11). ...

#### syn

East Los Angeles College, GENUS 0506
Excerpt: ... Linear Algebra - Dr Stoy - 14 MT + 8 HT Linear algebra pervades and is fundamental to geometry (from which it originally arose), algebra, analysis, applied mathematics, statistics-indeed all of mathematics. The course has several aims. The first is to introduce students through a thorough study of two- and three-dimensional spaces to the general concept of a vector space, subspaces, and the ideas of linear dependence, independence, spanning sets, bases, dimension. A second aim is to introduce students to matrices and their applications to the algorithmic solution of systems of linear equations and to the study of linear transformation s of vector spaces. A third aim is to introduce determinants and their properties. A fourth aim is to introduce eigenvalue theory and some of its applications. Fourteen lectures in Michaelmas Term Introduction: examples of linear problems (e.g., system of linear equations, differential equations) and their solutions. Vectors in the plane and 3-space ...

#### lec7

Excerpt: ... Math 3191 Applied Linear Algebra Lecture 7: Matrix Operations Stephen Billups University of Colorado at Denver Math 3191Applied Linear Algebra p.1/24 Announcements Hwk 4 and Study Guide 4 posted. ReminderExam 1 will be Sept. 27. Course Outline added to web site. Math 3191Applied Linear Algebra p.2/24 Outline Finish Sec. 1.9Matrix of a Linear Transformation . Sec. 2.1 Matrix Operations. Math 3191Applied Linear Algebra p.3/24 Sec 1.9: Matrix of a Linear Transformation Key Concepts: Identity Matrix and the vector ei . (Last time). Constructing the standard matrix of a linear transformation . Geometric linear transformation s in IR2 . one-to-one and onto. Math 3191Applied Linear Algebra p.4/24 Review ei is the vector with a '1' in the i-th component, and 0's everywhere else. x = x 1 e1 + x 2 e2 + + x n en . T (x) = T (x1 e1 + + xn en ) = x1 T (e1 ) + xn T (en ). Math 3191Applied Linear Algebra p.5/24 Finding the matrix of a linear transformation On the previous slide, no ...

#### 09S-60-Lecture12

Cal Poly Pomona, SG 064747
Excerpt: ... LECTURE 12: THE RANGE OF A LINEAR TRANSFORMATION MATH 60 SPRING 2009 Definition. Let A be an mn matrix and let TA : Rn Rm denote the associated linear transformation . The "input set" Rn is called the domain of TA and the "target set" Rm is called the codomain of TA . The range of TA is the set ran(TA ) = {TA (x) : x Rn } = {Ax : x Rn } = { y Rm : (x Rn )( Ax = y ) } of all possible outputs of TA . In other words, ran(TA ) = the set of all y Rm for which Ax = y is consistent. The symbol in the preceding definition means "there exists." It is important to note the distinction between the codomain (i.e., target set) and the range of a linear transformation : Although the range of a linear transformation is always a subset of the codomain, they are not necessarily equal. Consider the following example: Example 1. If T : R2 R2 denotes the orthogonal projection onto the x-axis. In other words, T is the linear operator on R2 whose standard matrix is A= In this case, we have T = TA whence domain(TA ) = R2 ...

#### Section5.1

CSU Fullerton, M 250
Excerpt: ... Math 250B 5.1 Linear Transformation s Summer 2007 Denition Let V and W be vector spaces. A mapping T from V into W is a rule that assigns to each vector v V exactly one vector w = T (v) in W . We denote such a mapping by T : V W . Denition Let V and W be vector spaces. A mapping T : V W is called a linear transformation from V to W if it satises the following: 1. T (u + v) = T (u) + T (v) for all u, v V . 2. T (cu) = c T (u) for all u V and all scalars c. A mapping that does not satisfy (either of) the conditions in the above denition is called a non linear transformation . Note that the above denition implies 1. T (0) = 0 2. T (v) = T (v) 3. T (c1 v + c2 u) = c1 T (v) + c2 T (u) for all linear transformation s T : V W , u, v V , c1 , c2 scalars. Example Which of the following are linear transformation s? 1. T : R3 R2 dened by T (x) = Ax, where A = 1 0 1 . 2 3 1 2. T : M2 (R) M2 (R) dened by T (A) = AT . 3. T : M2 (R) R dened by T (A) ...

#### final-rev

Hope, MATH 232
Excerpt: ... Major topics for Math 232 final exam: Chapter 1: Matrices: solving linear systems of equations, matrix inverses, determinants (and properties of and connections among these) Chapter 2: Vector spaces and subspaces (examples, basic properties), linear independence, span, basis, dimension, linear combinations, nullspace, row space, column space, showing linear independence of functions via Wronskians Chapter 3: Solving 1st order DEs or IVPs by separation of variables, exactness, or integrating factors, modeling with first order DEs, numerical methods (Euler/Taylor/Runge-Kutta) Chapter 4: Solving higher order, constant coefficient linear DEs or IVPs (homogeneous or not). Methods of undetermined coefficients and variation of parameters. Applications/modeling. Chapter 5: Linear transformation s (basic definitions and properties), image, kernel, injection, surjection, bijection, isomorphism, finding matrices for linear transformation or a column vector representation of a vector with respect to various bases, transit ...

#### final_prep

Uni. Worcester, MA 2073
Excerpt: ... MA 2073 Final Preparation Guide The following is a listing of parts of the text that have been covered since the mid-term. With each part, Ive included a few sentences to highlight the major topics; however, please note that you will be responsible for all material in these parts, except where explicitly noted, and also for related material covered in class. Sec. 2.3 Linear independence, basis, and dimension. Know these topics, especially the latter two, which werent covered on the mid-term. Sec. 2.4 The four fundamental subspaces. Know what these subspaces are, how to determine them, and how they and their dimensions are related. Sec. 2.6 Linear transformation s. Know the denition of a linear transformation and how to determine whether a function is a linear transformation . Know the basic linear transformation s introduced in this section (rotations, reections, and projections onto a line) and how to represent them as matrices. Know how to represent a linear transformation as a matrix. Sec. 3.1 Or ...

#### Math60Spring09Schedule

Cal Poly Pomona, MATH 04747
Excerpt: ... Math 60-Rumbos Spring 2009 Tentative Schedule of Lectures and Examinations Date W F M W F M W F M W F M W F M W F M W F M W F M W F Jan. 21 Jan. 23 Jan. 26 Jan. 28 Jan. 30 Feb. 2 Feb. 4 Feb. 6 Feb. 9 Feb. 11 Feb. 13 Feb. 16 Feb. 18 Feb. 20 Feb. 23 Feb. 25 Feb. 27 Mar. 2 Mar. 4 Mar. 6 Mar. 9 Mar. 11 Mar. 13 Mar. 16 Mar. 18 Mar. 20 Topic Introduction: n-dimensional Euclidean space Linear space structure in Euclidean space Linear combinations and spans Linear independence Linear independence and bases More on bases On linear transformation s between Euclidean spaces Matrix representation of a linear transformation Matrix representation of a linear transformation (continued) Matrix algebra Matrix algebra (continued) Function spaces Spaces of polynomials Vector spaces Subspaces Subspaces (continued): Spans and generating sets Generating sets (continued): Linear independence and bases Bases and Dimension Review Exam 2 Linear transformation s The dimension theorem for linear transformation s Composition of linear t ...

#### 302Lesson12

BYU, MATH 302
Excerpt: ... Linear Transformation s Theorem Geometric Interpretation Matrix Representation of a Linear Transformation Example Theorem Example Theorem Example Theorem Example ...

#### Math221Lecture008BHandouts

UMBC, MATH 221
Excerpt: ... The Identity Matrix The Identity Matrix The n n matrix that has ones along the main diagonal and zeros elsewhere is the identity matrix, denoted In . It is the reduced echelon form of a square n n matrix with a pivot in every column. If x is a vector in Rn , then In x = x Clint Lee Math 221 Lecture 8: The Matrix of a Linear Transformation 2/19 The Columns of the Identity Matrix The Identity Matrix The columns of the identity matrix In are vectors in Rn referred to as e1 , e2 , . . . , en . The 3 3 identity matrix is 1 I3 = 0 0 Its columns are 1 0 0 e1 = 0 , e2 = 1 , e3 = 0 0 0 1 0 1 0 0 0 1 Clint Lee Math 221 Lecture 8: The Matrix of a Linear Transformation 3/19 The Matrix of a Linear Transformation Any linear transformation can be written as a matrix transformation. The matrix for a linear transformation can be found by determining the effect of the transformation on the columns e1 , e2 , . . . , en of the identity matrix. Problem 19 page 80 illustrates this ...

#### examoutline

Brookdale, MATH 2030
Excerpt: ... Math 2030 Final Exam Outline Thursday June 26 6pm to 9pm Location: LSC 206 Topics Lecture 8: Basis, standard basis vectors of Rn , example of arguing one span is contained in another, computing basis of a row space, computing basis of a column space, computing basis of null space, dimension, uniqueness of size of bases for a given vector space, rank = dim col space = dim row space, rank of A equals rank A transposed, nullity, rank theorem, examples of computing nullity, invertible matrix theorem part two, vectors are uniquely represented by linear combinations of a basis. Lecture 9: Functions, domain, codomain, image, injective, surjective, bijective. Know the denitions, but you do not have to know the results about sizes of sets. Lecture 10: Linear transformation s, standard matrix, linear transformatios of linear combinations, constructing a linear transformation knowing the images of the standard basis vectors, all linear transformation s come from matrices, composition, kernel, kernels of a t ...