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Thread: gaussian and linear indep

  1. #1
    n22 is offline
    May 2013

    gaussian and linear indep

    Hello smart people,
    I am having some concept difficulty.
    This is what I dont understand......

    Determine the column space of AAttachment 28813(see image above)
    Use Gaussian elimination to check whether or not an arbitrary vector x
    Please see the following picture insert.
    Attachment 28812corresponds to a plane in R^3.Any basis of R^3 contains three vectors. Thus the four columns of A do not form a basis.(so its not a basis because there should be 3 vectors instead of 4.)
    also that y =( insert pic-plz see above) is the cartesian equation of a plane in 3-dimensional space, and
    only those vectors corresponding to a point on the plane are in the column space. So there arevectors in R^3 that do not belong to Col(A). (I dont understand how ther are vectors that dont belong in R^3?), Col(A) does not span R ..and hence is nota basis.(Since Col(A) is spanned by only two vectors, at most two of the columns of A form a linearly
    independent set.)
    Attached Thumbnails Attached Thumbnails gaussian and linear indep-question-3.png   gaussian and linear indep-after-elimination.png  
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  2. #2
    MHF Contributor
    Dec 2012
    Athens, OH, USA

    Re: gaussian and linear indep

    Hi n22,
    The question is to give a geometric argument for the column space. The attachment repeats your algebraic argument, but then does give a geometric argument:

    gaussian and linear indep-mhflinearalgebra7.png
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