what do they mean when they say a single vector is linear dependent?
is there a single vector that we can say it is linear dependent in R2?

Hi dingdong22!!

First of all see the following definition:
"A set D of vectors (in any vector space) is called dependent, if
there is at least one vector in D, that is a linear combination of the
other vectors of D.
A set of one vector is called dependent, if and only if it is the vector 0." 
From JCT-Jerusalem College of Technology, Israel website:
http://www.jct.ac.il/science/math/MathAbundance/vect.htm#14


That means set D ={D1,D2,...,Dj,...,Dn} is linear dependent if and
only if there is a vector Dj of D and a suitable set of real numbers
r,s,t,...,z , not all zero, such that r.D1+s.D2+t.D3+...+z.Dn = Dj .

So when somebody says that a single vector is linear dependent you must ask:
"Refered to which set of vectors?"

In regard to your second question if there is a single vector that we
can say it is linear dependent in R2, the definition is clear:
"A set of one vector is called dependent, if and only if it is the vector 0."

For more references visit the following pages:
"Linear dependent vectors" - JCT-Jerusalem College of Technology -
http://www.jct.ac.il/science/math/MathAbundance/vect.htm#14

"Determining if a set of Vectors is Linear Independent or Dependent"
By Benjamin T. Mueller - University of Wisconsin-Whitewater -
http://students.uww.edu/muellerbt15/LI.htm
 

I hope this helps you.
If you need more help, or find something obscure in my explanation, or
feel that the point is missed just use the Clarification feature to
ask me for further assistance.

Best regards and happy new year!!!
livioflores-ga

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Request for Answer Clarification by dingdong22-ga on 28 Dec 2003 09:20 PST

so the only single vector that is lineary dependent is vector 0,
because only vector zero satisfys av=0 that has non-trivial solutions?

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Clarification of Answer by livioflores-ga on 29 Dec 2003 06:23 PST

Hi!!

You said:
"so the only single vector that is lineary dependent is vector 0,
because only vector zero satisfys av=0 that has non-trivial solutions?"

The answer of this is yes and no.
Yes, vector zero satisfies av=0 that has non-trivial solutions.
But no, this is not the cause of it is the the only single vector set
that is lineary dependent is D={vector 0}.
A set of vectors is linearly dependent if at least one vector in the
set is a linear combination of the OTHER vectors. This is aplicable
only to set of two or more vectors. With a single vector set there are
not other vector to do a linear combination, then by convention (but
keeping the coherence) a set of one vector is linearly dependent if
and only if the vector is the zero vector and a set consisting of a
single non-zero vector is linearly independent; on the other hand, any
set containing the vector 0 is linearly dependent.

I hope this helps you.

Best regards.
livioflores-ga

Hi, dingdong22-ga:

To echo the point that I think livioflores-ga has well documented
above, to speak of a single vector being "linearly dependent" would
normally make sense only in relationship to some _set_ of vectors,
i.e. vector v is linearly dependent on set of vectors S.  One might
equally say v is in the span of S.

The more common usage of "linearly dependent" describes a set of
vectors, per the definitions cited by livioflores-ga.  The only
singleton set of vectors which is linearly dependent is the set
containing the zero vector {0}.  By convention the zero vector can be
expressed as an empty sum, and thus the zero vector is "linearly
dependent" upon an empty set of vectors.

Any other singleton set of vectors {v} is "linearly independent", as
the only solution for:

a v = 0

with scalar a and nonzero vector v, is a = 0.  [Note that in a vector
space the scalars are required to be elements of a field.  If a were
nonzero, then we could multiply both sides by the reciprocal of a,
obtaining v = 0 as a contradiction.]

regards, mathtalk-ga