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...