# gaussian and linear indep

• Jul 13th 2013, 05:22 AM
n22
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.)
• Jul 13th 2013, 06:04 PM
johng
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:

Attachment 28822