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Linearly independent vectors and span

  1. Nov 4, 2008 #1
    So the question is...

    Let v1, v2,...,,vn be linearly independent vectors in a vector space V. Show that v2,...,vn cannot span V.

    I honestly have found myself completely lost lately and I suck at writing proofs.

    So this is what I see,

    v1, v2,..., vn is linearly independent iff c1v1+c2v2+ ... + cnvn = 0 and c1=c2=...=cn= 0

    however I don't see why taking away v1 would make it not span V anymore.
  2. jcsd
  3. Nov 4, 2008 #2
    Consider this. How do you write v1 as a linear combination of v2, ..., vn?

    Or for a concrete example, in R^3. Take three vectors at right angles called x, y, and z. Can you write x as a linear combination of y and z? Obviously not, but go back to your definition of "span" and "linearly independent" and figure out why!
  4. Nov 4, 2008 #3


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    Do the following:

    1. Re-state the definition of linear independence carefully. (Your language was unclear.)
    2. Assume that you can express v1 as a linear combination of the other vectors.
    3. Show that this will contradict linear independence.
    4. The conclusion is that you cannot express v1 using the other vectors.

    It's very important to know your definitions precisely!
  5. Nov 4, 2008 #4
    In an n-dimensional vector space, a linearly independent set of n vectors is a basis for that space. Also, in an n-dimensional vector space, every basis for that space has the same amount of vectors. A basis is a linearly independent set that spans the space. This makes the rest of the proof easy.
    Last edited: Nov 4, 2008
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