Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Basis of a nullspace

  1. Oct 13, 2009 #1
    I am in a linear algebra and differential equations course and have recently been learning how to find a basis for a nullspace, row space, or column space. However, I am EXTREMELY confused by a solution to a question in my textbook. The question asks to find the basis for the null space of a particular matrix. For the question at hand, the determinant is zero, which of course means that it is invertible. However, I'm confused as to why this means that there is no basis for this nullspace -- there is no relevant explanation in my text as far as I can tell. Thank you to anyone who can help solve this problem for me!
     
  2. jcsd
  3. Oct 13, 2009 #2
    If the determinant is zero then the matrix is not invertible
     
  4. Oct 13, 2009 #3
    I'm sorry, I meant to write not zero....my mistake; that makes a big difference. :)
     
  5. Oct 13, 2009 #4
    This all stems from the fact that matrices distribute over vectors and commute with scalar multiplication: i.e, the fact that matrices are linear transformations.

    If Av=0 and Ax=r then A(v + x) = Av + Ax = 0 + r = r.

    If I have one solution to Ax = r, then I also have a solution for every element of the nullspace. In this situation, how would I write A-1? Is it A-1r = x or A-1r = x + v? They both get sent to r by A.

    This means that a matrix can only be invertible if the nullspace is trivial, in other words, if only zero gets sent to zero.
     
  6. Oct 14, 2009 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    "Nullspace of A" is, by definition, the set of vectors, x, such that Ax= 0. If A has an inverse, take [itex]A^{-1}[/itex] on both sides: [itex]A^{-1}Ax= x= A^{-1}0= 0[/itex].

    That is, if A is invertible, Ax= 0 only for x= 0.

    It is true, then that if A is invertible, it is one-to-one.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook