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!

Linear maps (rank and nullity)

  1. Jan 18, 2012 #1
    1. The problem statement, all variables and given/known data
    A linear map T: ℝ[itex]^{m}[/itex]-> R[itex]^{n}[/itex] has rank k. State the value of the nullity of T.

    3. The attempt at a solution

    I know that rank would be the number of leading columns in a reduced form and the nullity would simply be the number of non-leading columns, or total columns - rank.

    As we are not given how many different vectors are in the matrix, I'm not too sure how to give a value.
     
  2. jcsd
  3. Jan 18, 2012 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    The rank is given.
    What is the total number of columns for a map [itex]\mathbb{R}^m \to \mathbb{R}^n[/itex]?
     
  4. Jan 19, 2012 #3
    I'm inclined to say m, but I'm not entirely sure. So m - k? Considering that's the starting space if you will.
     
  5. Jan 19, 2012 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The "rank-nullity theorem" which, if you were given this problem, you are probably expected to know, says that if A is a linear transformation from a vector space of degree m to a vectoer space of degree n then rank(A)+ nullity(A)= m.
     
  6. Jan 19, 2012 #5

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    If it's a map from [itex]\mathbb{R}^m[/itex] to [itex]\mathbb{R}^n[/itex], that means that you should be able to multiply the matrix with a vector with m components, and this should give a vector with n components. If you think about the way matrix multiplication works, does the number of columns need to be m or n?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Linear maps (rank and nullity)
  1. Rank and nullity (Replies: 7)

  2. Rank and nullity (Replies: 3)

Loading...