Question

set of vectors {v1, v2.......,vn} belongs to R^m where n 13 years ago

Answer 1

not necesarily. for instance
\[ \large \{(1,2,3),(0,0,0)\} \]
is a linearly dependent set of two vectors in \(\mathbb{R}^3\) and \(2<3\).

Answer 2

how to prove this?

Answer 3

since you can prove that, for at least one case this is not true, therefore it can not be generalized.

Answer 4

To prove that (1,2,3) and (0,0,0) are dependent:
Let us assume the contrary. If they are independent, then
k1* (1,2,3) + k2 * (0,0,0) = (0,0,0) implies k1 = k2 = 0.
But, in this case, only k1 has to be 0. k2 can be any number and the expression still holds. Hence the proof.