1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Column space

  1. Jun 19, 2007 #1

    413

    User Avatar

    How would I prove this theorem:

    "The column space of an m x n matrix A is a subspace of R^m"

    by using this definition: A subspace of a vector space V is a subset H of V that has three properties: a) the zero vector of V is in H. b) H is closed under vector addition. c) H is closed under multiplication by scalars.

    Please help
     
  2. jcsd
  3. Jun 20, 2007 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Okay, here's what you should do:
    1. Write out the definition of "column space" since you already have the definition of "subspace".

    2. Show that the column space is a subset of R^m.

    3. Show that (a) is true: is the zero vector in the column space- does it satisfy the definition of vectors in the column space?

    4. Show that (b) is true: if you add two vectors in the column space is the result in the column space?

    5. Show that (c) is true: if you multiply a vector in the column space by a scalar is the result in the column space?
     
  4. Jun 21, 2007 #3

    413

    User Avatar

    I think i can show the three properties, but how would i show the column space is a subset of R^m?
     
  5. Jun 21, 2007 #4

    radou

    User Avatar
    Homework Helper

    Well, it's obvious since all the columns are from R^m !
     
  6. Jun 23, 2007 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Again, what is the definition of "column space"?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Column space
  1. Column Spaces (Replies: 1)

Loading...