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Linear Algebra-Linear dependence

  1. May 24, 2014 #1
    1. The problem statement, all variables and given/known data

    let U be a 3x3 matrix containng columns C1, C2, C3. The three column vectors C1= (a,0,0) , C2=(b,d,0), C3=(c,e,f)
    prove that if a=0 or d=0 or f=0 (3cases), the columns of U are dependent?
    problem from Linear algebra and applications, fourth editon, Gilbert strang


    2. Relevant equations
    no eqations


    3. The attempt at a solution
    I successfully proved the first two cases
    if a=0, if we multiply C3 or C2 with zero then C1 will be equal to C2 or C3. The columns become independent
    if d=0, if we multiply C1 with b/a and C2 with a/b, C1 and C2 will be identical and the columns become independent.
    But i dont know how to prove the third case, i tried with different comibinations of scalars with multiply with C2 and C3, but i can't make this two columns identical. Enlighten me.
     
  2. jcsd
  3. May 24, 2014 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The third case has f= 0 so the three vectors are (a,0,0), (b, d, 0), and (c, e, 0).
    You want to find numbers x and y such that (c, e, 0)= x(a, 0, 0)+ y(b, d, 0).
    That is equivalent to the two equations c= xa+ yb and e= yd. Obviously, if d is not 0, y= e/d and your first equation become c= xa+ be/d. It should be easy to solve that for x. If d= 0 there are an infinite number of soutions.
     
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