MATH 10 Linear Algebra And Applications

• 7 pages

  • Test 1 SOLUTIONS.pdf
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2014
  • Math 10 Solutions to Test 1 Page 1 Page 2 Page 3 Page 4 6. 7. 8. 9. Since B is obtained from A by doing the following ERO on A: 2 R3 + R1 R1 We have that the elementary matrix E such that EA = B is obtained from I by doing the same ERO: 1 0 0 1 0 2 + R
  • Register Now

  Test 1 SOLUTIONS.pdf

• 33 pages

  • jtsocrates_36994
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • Chapter Zero The Language of Mathematics: Sets, Axioms, Theorems & Proofs Mathematics is a language, and Logic is its grammar. Chapter Zero: The Language of Mathematics 1 Part I: Set Theory and Basic Logic Definition: A set is an unordered collection of o
  • Register Now

  jtsocrates_36994

• 8 pages

  • jtsocrates_41240
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 3.2 Linearity Properties for Finite Sets of Vectors Linear Combinations and Spans of Finite Sets Definition: Let S 1 , 2 , . . . , n be a set of vectors from a v v v vector space V, , , and let r 1 , r 2 , . . . , r n . Then, a linear combination of the v
  • Register Now

  jtsocrates_41240

• 10 pages

  • jtsocrates_41291
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 3.6 Coordinate Vectors and Matrices for Linear Transformations Definition: Let B w 1 , w 2 , . . . , w n be an ordered basis for a finite dimensional vector space V. If is any vector in V, we know v that can be expressed uniquely as a linear combination
  • Register Now

  jtsocrates_41291

• 8 pages

  • jtsocrates_41290
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 3.5 Linear Transformations on General Vector Spaces Definition: A linear transformation: T : V, V , V W, W , W is a function that assigns a unique member w W to every vector V, such that T satisfies for all , V and all scalars v u v k: The Additivity Pr
  • Register Now

  jtsocrates_41290

• 13 pages

  • jtsocrates_41269
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 3.4 Subspaces, Basis and Dimension Subspaces Definition: A non-empty subset W of a vector space V, , is called a subspace of V if W is closed under and . In other words, for all w 1 and w 2 W, and k : w 1 w 2 W, and k w 1 W. As before, we write W V, and w
  • Register Now

  jtsocrates_41269

• 7 pages

  • jtsocrates_41455
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • Section 3.8 Isomorphisms Definition: We say that T : V W is an isomorphism if T is both one-to-one and onto. We also say that T is invertible, T is bijective, and the vector space V is isomorphic to W. In the case when V W, we call T an automorphism of V.
  • Register Now

  jtsocrates_41455

• 13 pages

  • jtsocrates_41457
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 3.7 One-to-One and Onto Linear Transformations ; Compositions of Linear Transformations Review: . V . 0. . W . V V T . . . 0 . . V 0W ker (T) . . . T W .0 . . . . . . W . range ( T ) . kerT rangeT Section 3.7 V | T 0 W v v w W | w T for some V v v 1 On
  • Register Now

  jtsocrates_41457

• 2 pages

  • Groupwork 5 Solutions.pdf
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2014
  • Groupwork 5 Solutions Section 4.2 #14 (a) cos 2 x spancfw_f, g Q cos 2 x = cos 2 x sin 2 x = 1f 1g (b) 3 + x 2 spancfw_f, g (c) 1 spancfw_f, g Q 1 = sin 2 x + cos 2 x = 1f + 1g (d) sin x spancfw_f, g (e) 0 spancfw_f, g Q 0 = 0 sin 2 x + 0 cos 2 x = 0f + 0
  • Register Now

  Groupwork 5 Solutions.pdf

• 4 pages

  • Groupwork 3 Solutions.pdf
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2014
  • Groupwork 3 Solutions Section 1.7 #29 If A is an invertible upper triangular or lower triangular matrix, then the diagonal entries of A 1 are the reciprocal of the ones of A. Proof: Let A be an n n invertible upper triangular matrix, then A 1 is also uppe
  • Register Now

  Groupwork 3 Solutions.pdf

• 1 pages

  • Groupwork 6 Solutions PART 2.pdf
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2014
  • Groupwork 6 Solutions Chapter 4 Supplementary Exercises (page 289) 7. Under the given hypotheses in the problem, we claim that if A is invertible, then Av1 , Av 2 , K , Av n is a linearly independent set. Proof of the claim: We want to show that the vecto
  • Register Now

  Groupwork 6 Solutions PART 2.pdf

• 3 pages

  • Groupwork 6 Solutions PART 1.pdf
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2014
  • Groupwork 6 Solutions Chapter 4 Supplementary Exercises (page 289) 4. (a) (4a, a b, a + 2b) = a (4, 1, 1) + b(0, 1, 2) (b) (3a + b + 3c, a + 4b c, 2a + b + 2c) = a (3, 1, 2) + b(1, 4, 1) + c(3, 1, 2) = (a + c)(3, 1, 2) + b(1, 4, 1) (c) (2a b + 4c, 3a c, 4
  • Register Now

  Groupwork 6 Solutions PART 1.pdf

• 2 pages

  • Practice Problems for Final.pdf
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2014
  • Problem 1 (15 points). Determine h. and L‘ such that the solution set of 1‘1 + 31‘? = it 411 + hr? = 8 (a) is empty, (b) contains a unique solution and (:2) contains infinitely many solutions. (Give separate answers for each part. andjnstfljv them.) Problem
  • Register Now

  Practice Problems for Final.pdf

• 12 pages

  • jtsocrates_41217
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 3.1 Axioms for a Vector Space Definition (The Axioms of an Abstract Vector Space): A vector space V, , is a non-empty set V, together with two operations: vector addition, and scalar multiplication, such that: for all , and w V and all r, s , u v V, , s
  • Register Now

  jtsocrates_41217

• 7 pages

  • jtsocrates_41096
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 2.8 Consequences of Invertibility Theorem (The Really Big Theorem on Invertibility): The following conditions are equivalent for a linear operator T : n n , with standard matrix T A : 1. T is an invertible operator. 2. A is an invertible matrix. 3. The rr
  • Register Now

  jtsocrates_41096

• 22 pages

  • jtsocrates_41016
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 1.7 Subspaces of Euclidean Spaces Definition: A subspace W of n is a non-empty subset of vectors of n such that if , W, and r , then we also have: u v W and r W. u v v We say W is closed under vector addition and scalar multiplication, and write: W n to
  • Register Now

  jtsocrates_41016

• 19 pages

  • jtsocrates_36992
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 1.2 The Span of a Set of Vectors Definition: The Span of a non-empty set of vectors S 1 , 2 , . . . , k from n is the set of all possible linear v v v combinations of the vectors in the set. We write: SpanS Span 1 , 2 , . . . , k v v v x 1 1 x 2 2 x k k
  • Register Now

  jtsocrates_36992

• 21 pages

  • jtsocrates_36991
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • Chapter 1 The Canvas of Linear Algebra: Euclidean Spaces and Subspaces 1 1.1 The Main Subject: Euclidean Spaces Definition: An ordered n tuple or vector is an ordered list of n real numbers: v 1 , v 2 , . . . , v n . v Definition: The set of all possible
  • Register Now

  jtsocrates_36991

• 15 pages

  • jtsocrates_40983
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 1.6 Independent Sets versus Spanning Sets Theorems connecting Spans of sets of vectors, and linearly independent or dependent sets. The concepts of Spans and independence are two of the most important concepts in Linear Algebra. Section 1.6 Independent Se
  • Register Now

  jtsocrates_40983

• 24 pages

  • jtsocrates_40979
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 1.5 Linear Systems and Linear Independence Definition: A linear system is called consistent if it has at least one solution. A system is called inconsistent if it does not have any solutions. Theorem: Let m and let S 1 , 2 , . . . , n be a set of b v v v
  • Register Now

  jtsocrates_40979

• 12 pages

  • jtsocrates_41088
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 2.1 Mapping Spaces: Introduction to Linear Transformations Definition: Let X and Y be any two sets. A function f : X Y is a rule (or a recipe, or a formula) that receives as its input an element x of X, and assigns to x as its output a unique element y of
  • Register Now

  jtsocrates_41088

• 6 pages

  • jtsocrates_41090
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 2.3 Operations on Linear Transformations and Matrices Definitions: If T 1 : n m and T 2 : n m are linear transformations, and k , then we can define: T1 T2 : n m, T 1 T 2 : n m , and kT 1 : n m , as linear transformations, respectively, by: with actions g
  • Register Now

  jtsocrates_41090

• 12 pages

  • jtsocrates_41089
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 2.2 Rotations, Projections and Reflections Rotations in 2 . rot ( v ) r r rot ( v ) v r + . v A Vector and rot , v its Counterclockwise Rotation by Theorem: The function rot : 2 2 that takes a vector and rotates counterclockwise by an angle of v v about
  • Register Now

  jtsocrates_41089

• 18 pages

  • jtsocrates_41026
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 1.8 Basis and Dimension Definition: A basis for a non-zero subspace W n is a non-empty set of vectors B w 1 , w 2 , , w k which Spans W and is also linearly independent. Since SpanB includes the vectors w 1 , w 2 , . . . , w k , these vectors automati
  • Register Now

  jtsocrates_41026

• 13 pages

  • jtsocrates_41091
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 2.4 Properties of Operations on Linear Transformations and Matrices Goal: Show that matrix operations enjoy many (but not all!) of the properties of the analogous operations on ordinary real numbers. Section 2.4 Properties of Operations on Linear Transfor
  • Register Now

  jtsocrates_41091

• 11 pages

  • jtsocrates_41092
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 2.5 One-to-One Transformations and Onto Transformations Preservation of Subspaces Theorem (The Preservation of Subspaces Theorem): Suppose that T : n m is a linear transformation. Then, for any subspace V of n , and any subspace W of m : v TV w m | w T fo
  • Register Now

  jtsocrates_41092

• 11 pages

  • jtsocrates_41095
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 2.7 Finding the Inverse of a Matrix Goal: to be able to construct the matrix of the inverse of an invertible linear operator, and at the same time, to find the inverse of an invertible square matrix which is 3 3 or bigger, when it is possible to do so. Se
  • Register Now

  jtsocrates_41095

• 11 pages

  • jtsocrates_41094
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2015
  • 2.6 Invertible Operators and Matrices In Algebra, we first require that a function: f : D R, with domain D and range R, is one-to-one on D before we find its inverse. If so: f 1 : R D, where f 1 y x if and only if f x y. f and f 1 also possess the cancell
  • Register Now

  jtsocrates_41094

• 5 pages

  • Practice Problems for Final SOLUTIONS.pdf
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2014
  • Problem 1 (15 points). Determine h and I; such that the solution set of $14-31": = 4331+ Il'tl'j = 8 {a} is empty. {b} contains a unique solution. and {c} contains infinitely many solutions. {Give separate answers for each part. and justify them.) Solutio
  • Register Now

  Practice Problems for Final SOLUTIONS.pdf

• 4 pages

  • Test 2 SOLUTIONS.pdf
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2014
  • 1. 2. 3. (Procedure may vary; sample answer) 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Since ( A + AT )T = AT + (AT ) = A + AT , A + AT is symmetric. T 15.
  • Register Now

  Test 2 SOLUTIONS.pdf

Recent Documents

• 7 pages

  • Armstrong Homework 6.pdf
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2008
  • Name: STATS 67, Spring 2017 Homework 6 David Armstrong Due May 26, 2017 Please include your R code if used to calculate probabilities. 1 1. If the size of a sample randomly selected sample from a population is increased from n = 100 to n = 400, then the s
  • Register Now

  Armstrong Homework 6.pdf

• 9 pages

  • Armstrong Homework 5.pdf
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2008
  • Name: STATS 67, Spring 2017 Homework 5 David Armstrong Due May 19, 2017 Please include your R code if used to calculate probabilities. 1 1. Suppose X1 , . . . , Xn is a set of n observations representing the Bernoulli probability model. (a) Write out the
  • Register Now

  Armstrong Homework 5.pdf

• 5 pages

  • chapter_5.doc
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2008
  • Chapter 5- Marriage and the Politics of Yams YAMS AS WEALTH - - - - During every harvest, the gardener gives yams to the owner (woman),which are carried to the owners hamlet, dismantled and put into woven baskets. Young people come to the garden to carry
  • Register Now

  chapter_5.doc

• 7 pages

  • anthro immigration report.docx
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2008
  • Eric Kerikorian Professor Douglas Anthropology 2A 05/13/17 Course Paper: Immigration Numerous different aspects may contribute to an individuals decision to emigrate from their home country. Push factors are defined in the dictionary as any negative aspec
  • Register Now

  anthro immigration report.docx

• 1 pages

  • ekerikor_Vim Commands-Macros.pdf
  • Pasadena City College
  • Linear Algebra And Applications
  • MATH 10 - Spring 2008
  • Eric Kerikorian Professor Klefstad ICS 53 05/09/17 HW 5 Question 2 a) 3w-cw-an idiot b) ESC-ma-^-i-IF c) ESC-`a-e-a- savant d) ESC-0-l-rf e) O-ESC-ZZ f) :map! \fo <Tab>for(>;>;>)<Enter><Tab>cfw_<Enter><Tab><Tab><Enter><Tab><Esc>/;<Enter>i g):map !S /<Tab>
  • Register Now

  ekerikor_Vim Commands-Macros.pdf