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Linear Algebra - Linear (in)dependence of a set

  1. May 27, 2017 #1
    1. The problem statement, all variables and given/known data

    Let { u, v, w} be a set of vectors linearly independent on a vector space V

    - Is { u-v, v-w, u-w} linearly dependent or independent?

    2. Relevant equations

    A set of vectors u, v, w are linearly independent if for the equation

    au + bv + cw= 0 (where a, b, c are real scalars)

    The only solution is the trivial one a = b = c = 0


    3. The attempt at a solution

    I expressed the vector in base of their coefficients, changed them into column vectors, formed a matrix and solve the homogeneous system

    (1)u - (1)v + (0)w
    (0)u + (1)v - (1)w
    (1)u + (0)v - (1)w

    | 1 0 1 |
    |-1 1 0 |
    | 0 -1 -1 |

    After doing (R2 + R1 on R2) > (R3 + R2 on R3) I get:

    | 1 0 1|
    | 0 1 1|
    | 0 0 0|

    Which has no pivot on the 3rd column, thus the system has multiple solutions, making the set linearly dependent, however my answer key says is linearly independent and I'm not sure what I may be doing wrong.
     
  2. jcsd
  3. May 27, 2017 #2

    vela

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    Nothing. The key is wrong. If you add the first and second vectors together, you get the third. That set is clearly dependent.
     
  4. May 27, 2017 #3
    Thank you, I didn't notice that indeed the 3rd vector is the sum of the first two, that is proof enough the key is indeed wrong.
     
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