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Prove coordinate vectors are unique for given basis

  1. Dec 15, 2011 #1
    1. The problem statement, all variables and given/known data
    Prove that the coordinates of a vector v in a vector space Vn are unique with respect to a given basis B={b1,b2,...,bn}


    2. Relevant equations



    3. The attempt at a solution
    not sure at all what to do with this
     
  2. jcsd
  3. Dec 15, 2011 #2

    radou

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    Homework Helper

    OK, this is very straightforward. Assume you can represent a vector v as two linear combinations of the basis vectors with different coefficients. After that you only need to use a fundamental property of the basis vectors and that's it.
     
  4. Dec 15, 2011 #3
    Well you want to show that it's unique. A strategy that's good for this kind of proof is to assume the opposite, work with that assumption, and then arrive at something that shows that it must be unique.

    To get started:
    Well what's the opposite of being unique? How about we assume that a vector can be represented by this basis in two different ways as a linear combination but with different constants. What do you think you can do with this? Think about how you could manipulate this to show that the constants must be the same. You'll need to know properties of a basis to make it work.
     
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