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Vector is not a linear combination of two vectors [matrix algebra]

  1. Feb 16, 2010 #1
    1. The problem statement, all variables and given/known data
    Find a vector w in R3 such that w is not a linear combination of v1 and v2


    2. Relevant equations
    v1 = [1;2;-1] v2=[2;-1;-2]


    3. The attempt at a solution

    my question is : does w need to be in the span of v1 and v2?? Could i just choose [1; 1; 1] as my w vector??
     
  2. jcsd
  3. Feb 16, 2010 #2

    Matterwave

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    If w is NOT a linear combination of v1 and v2 then it CANNOT be in the span of v1 and v2.

    An easy way of finding one such vector is to take the cross product of v1 and v2. The cross product of v1 and v2 is orthogonal to both v1 and v2 and is therefore not within the span of v1 and v2. But really, any vector not spanned by v1 and v2 is perfectly acceptable.
     
  4. Feb 16, 2010 #3
    that makes perfect sense! thank you
     
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