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LA: basis for a subspace

  1. Oct 5, 2008 #1
    1. The problem statement, all variables and given/known data

    Find a basis for the subspace S of R^4 consisting of all vectors of the form

    (a+b, a-b+2c, b, c)^T, where a,b,c are real numbers. What is the dimension of S?

    2. Relevant equations

    vectors v1,...,vn from a basis for a vector space iff

    i) v1,...,vn are linearly independent
    ii) v1,...,vn span V

    3. The attempt at a solution

    I'm actually not sure how I'd start this problem as most of the basis problems I've been doing have been comparing 2-3 vectors against each other not just one. Would I have to find a vector of scalars in R^4 and find S is both linearly independent and spans R^4 against it? any help is greatly appreciated.
     
  2. jcsd
  3. Oct 5, 2008 #2

    statdad

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    Homework Helper

    Try writing the given vector (the one in your post) as a sum of several, with [tex] a, b, c [/tex] as the coefficients of the linear combination. Are the vectors you find linearly independent?
     
  4. Oct 5, 2008 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    In particular, the zero vector, here (0,0,0,0)T, is in every subspace. For what values of a, b, c is (a+b, a-b+2c, b, c)T equal to the vector?
     
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