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Homework Help: Linear algebra

  1. Jan 29, 2009 #1
    i am given a subspace of R4 W={(a b c d)} and know a+c=0 c-2d=0, and am asked to find a basis for W,
    i wrote (-2 0 2 1)(0 1 0 0),
    now i am asked to find the missing vectors so that the new basis will be a basis for R4. to find this i need vectors that are independant and are the basis for R4-W???? how can i do this??
    just a guess would be (-2 0 2 1)(0 1 0 0)(0 0 1 0)(0 0 0 1) but thats just because i know that those four would cover the whole space and are independant. what is the mathematical way of solving this?
     
  2. jcsd
  3. Jan 29, 2009 #2

    Mark44

    Staff: Mentor

    Gram-Schmidt process --http://en.wikipedia.org/wiki/Gram_schmidt
     
  4. Jan 29, 2009 #3
    another question, not related,
    is it possible for 2 seperate basises of a subspace to be linearly independant of one another, or do they always need to be dependant
     
  5. Jan 29, 2009 #4

    Mark44

    Staff: Mentor

    I don't think it's meaningful to talk about one basis being independent of another. For linear independence/dependence, we're always talking about a collection of vectors.

    Suppose v1, v2, ... , vn are a basis for a subspace W and u1, u2, ..., un are another basis for W. The set of vectors {v1, v2, ..., vn, u1} has to be a linearly dependent set, meaning that u1 has to be a linear combination of v1, v2, ... , vn.

    A basis for a subspace W is the largest set of vectors that a) is linearly independent, and b) spans W. If you add any vector to this basis, the added vector must be a linear combination of the original basis vectors.
     
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