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More linear algebra

  1. Apr 5, 2006 #1
    Suppose [itex] X_{1},X_{2},...,X_{N} [/itex] ae vectors in Rn. If [itex] Y = a_{1} X_{1} ... + a_{N} X_{N} [/itex] where ai is not zero, show that
    [tex] span{X_{1},...,X{N}} = span{Y,X_{2},...,X_{N}} [/tex]

    [tex] span{X_{1},...,X{N}} = a_{1} X_{1} + ... + a_{N} X_{N} [/tex]
    [tex] Y = a_{1} X_{1} + ... + a_{N} X_{N} [/tex]
    then [tex] bX_{1} = Y - a_{2} X_{2} ... - a_{N} X_{N} [/tex]

    so i can see that [tex] bX_{1} = span{Y,X_{2},...,X_{N}} [/tex]
    also we know that X 1 has a unique representation as a span of the Xi, where i is not 1

    but i m not sure how connect the two...
  2. jcsd
  3. Apr 5, 2006 #2


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    Your notation is wrong. Span x1, ... xn is not equal to a1x1 ... anxn. Instead, you know that if v is an ELEMENT of Span x1, ... xn, then v can be written as a1x1 ... anxn with not all of the ai's zero.

    You are trying to show that a vector v is in Span x1, ... xn, if and only if it is in span y, x2, ... xn.
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