a question about Linear Algebra ---20 POINTS!

其實不難
Definition:
A set
圖片參考：http://distance-ed.math.tamu.edu/Math640/chapter3/img171.gif
is called linearly independent if the only solution to the equation


圖片參考：http://distance-ed.math.tamu.edu/Math640/chapter3/img601.gif


is
圖片參考：http://distance-ed.math.tamu.edu/Math640/chapter3/img602.gif
. If the set is not linearly independent, it is called linearly dependent.
from the question, {v1,v2,v3} be a linearly dependent set
So there exist c1,c2,c3, not all 0 such that
c1v1+c2v2+c3v3=0
Now if we let c4=0, then
c1v1+c2v2+c3v3+c4v4=0 (since c4v4=0*v4=0)
We have found c1,c2,c3,c4 (because at least one of c1,c2,c3 is not 0) not all 0 such that c1v1+c2v2+c3v3+c4v4=0. By the definition, we prove that the vectors {v1,v2,v3,v4} form a linearly dependent set .