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Homework Help: Basis of a vector space

  1. Nov 8, 2013 #1
    1. The problem statement, all variables and given/known data

    There is a standard basis, B = (1; z; z^2; z^3; z^4) where B is the basis of a R4[z] of real polynomials of at most degree 4.

    I need to find another basis B' for R4[z] such that no scalar multiple of an
    element in B appears as a basis vector in B' and also prove that this B' is a basis.

    Can any help with this please?

    2. Relevant equations

    3. The attempt at a solution

    I could only think to do a basis B'=(0,1,z,z^2,z^3) but no sure if this is correct.
  2. jcsd
  3. Nov 8, 2013 #2


    Staff: Mentor

    No, it isn't correct. The 0 vector can never be a vector in a basis for a vector space.

    Here's a hint: the function 1 + z is not a linear multiple of any of the vectors in B, right?
  4. Nov 8, 2013 #3
    So you could have B' = (1+z, 1+z^2, 1+z^3, 1+z^4, 1+z^5) for example or would this not work because the highest degree for R4[z] is 4?

    Thanks for your reply.
  5. Nov 8, 2013 #4


    Staff: Mentor

    Right, you can't have 1 + z5. Can't you think of any other combinations besides adding 1 to another vector in the basis?
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