1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Linear Algebra: Null Space and Dimension

  1. Apr 18, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that dim(nullA) = dim(null(AV))
    (A is a m x n matrix, V is a n x n matrix and is invertible

    2. Relevant equations

    AX=0 and AVX = 0
    Null(AV) = span{X1,..Xd}
    Null(A) = span{V-1X1,.., V-1Xd}

    3. The attempt at a solution

    so you need to prove that dim(null(AV) is a subset of nullA?

    Therefore d = dimA = dim(AV) and let nullA= span{X1,..Xd} and null(AV) = span{ X1,..Xd}.
    Null(AV) = span{X1,..Xd}
    Null(A) = span{V-1X1,.., V-1Xd}

    If a1(V-1X1 ) + …+a2(V-1Xd) = 0
    So 0 = VV-1(t1X1 +… + t1Xd)
    0 = t1X1 +… + t1Xd all of ti = 0 meaning it is linearly independent, and also span {V-1X1,.., V-1Xd} which is a basis of nullA

    AX = 0 so AVX=O then
    V0 = AVX which is in null(AV)
    This also means V0 = t1X1 +… + t1Xd
    which also means 0 = t1V-1X1 +…+ tdV-1Xd which spans nullA.

    Is this correct?

    Thank you, in advance :)
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted