Documents about Linear Transformation

  • 2 Pages


    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 ...

  • 2 Pages


    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 ...

  • 1 Pages


    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 ...

  • 4 Pages


    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:/ Here you will find a diagram, labelled "Part I," that allows you to ...

  • 2 Pages


    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 ...

  • 2 Pages


    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). ...

  • 3 Pages


    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 ...

  • 24 Pages


    University of Colorado Denver, MA 3191

    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 ...

  • 2 Pages


    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) ...

  • 2 Pages


    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 ...

  • 12 Pages


    BYU, MATH 302

    Excerpt: ... Linear Transformation s Theorem Geometric Interpretation Matrix Representation of a Linear Transformation Example Theorem Example Theorem Example Theorem Example ...

  • 1 Pages


    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 ...