{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter2 - Chapter 2 Vector Spaces Keith E Emmert Euclidean...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 2: Vector Spaces Keith E. Emmert Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Chapter 2: Vector Spaces Keith E. Emmert Tarleton State University January 14, 2010
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Chapter 2: Vector Spaces Keith E. Emmert Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Overview Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces
Image of page 2
Chapter 2: Vector Spaces Keith E. Emmert Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Common Notation Definition 1 N = { 1 , 2 , 3 , . . . } is the set of natural numbers . W = { 0 , 1 , 2 , 3 , . . . } is the set of whole numbers . Z = {− 3 , 2 , 1 , 0 , 1 , 2 , 3 , . . . } is the set of integers . Q = braceleftBig a b | a , b Z , b negationslash = 0 bracerightBig is the set of rational numbers . R is the set of real numbers . Let i = 1. Then C = { a + bi | a , b R } is the set of complex numbers . The set of polynomials whose degree is less than or equal to n is P n = { a 0 + a 1 t + a 2 t 2 + · · · + a n t n | a 0 , a 1 , . . . , a n R } . M m × n is the set of all (real valued) matrices of size m × n ( m rows and n columns).
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Chapter 2: Vector Spaces Keith E. Emmert Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Vector Space Defined Definition 2 Let set F be a set of elements, scalars , with operations , (addition & multiplication). A vector space is a set V of elements, vectors , with vector addition , , and scalar multiplication , , satisfying the ten axioms: Additive Properties 1. Closure for Addition If u , v V , then u v V . 2. Commutativity of Addition If u , v V , then u v = v u . 3. Associativity of Addition If u , v , w V , then ( u v ) w = u ( v w ) . 4. Existence of a Zero Vector There exists 0 V such that for all u V , u 0 = u . 5. Existence of Additive Inverses For all u V , there exists at least one element ˜ u V such that ˜ u u = 0 . Fact: ˜ u is unique! Let ˜ u = u .
Image of page 4
Chapter 2: Vector Spaces Keith E. Emmert Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Vector Space Definition Continued Definition 2 (Continuation of Vector Space Definition) Product Properties 1. Closure for Scalar–Vector Product If α F and u V , then α u V . 2. Distributive Law: Scalar Times Vectors For any α F and u , v V , we have α ( u v )=( α u ) ( α v ) . 3. Distributive Law: Scalar Sum Times Vector For any α, β F and u V , we have ( α β ) u =( α u ) ( β u ) . 4. Associativity of Scalar–Vector Product If α, β F and u V , then α ( β u )=( α β ) u . 5. Multiplicative Identity: Unit Scalar Times Vector There exists 1 F such that for each u V , 1 u = u .
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Chapter 2: Vector Spaces Keith E. Emmert Euclidean Vector Spaces Lines, Planes, and Hyperplanes Linear Transformations General Vector Spaces Remarks Remark 3 Before you talk about a vector space, you must define
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern