# Linearly Dependent Vectors

n Vectors v_1, v_2, .., v_n, are linearly dependent iff there exist scalars c_1, c_2, .., c_n (not all zero), such that

c_1*v_1 + c_2*v_2 + ... + c_n*v_n = 0.

By this definition, it would seem that the following vectors are linearly dependent:

v_1=(0,0,0), v_2, v_3, .., v_n

(For any c_1 and c_i=0 (for 1<i<=n))

However, this seems (to me) to be on the same par as saying "0 is a multiple of 5"...

So, is there some technicality which resolves this? Are the above linearly dependent or not?

Thanks
LVL 25
###### Who is Participating?

Commented:
Determinant of
| 0 v21 v31 |
| 0 v22 v32 |
| 0 v23 v33 |
is 0
0

Commented:
v_1=(0,0,0) is not independent of v_2, v_3
0

Commented:
>> By this definition, it would seem that the following vectors are linearly dependent:
>> v_1=(0,0,0), v_2, v_3, .., v_n
>> (For any c_1 and c_i=0 (for 1<i<=n))

Yes, this is correct. These vectors are linearly dependent.
Because v_1 is a zero vector, equation:

c_1*v_1 + c_2*v_2 + ... + c_n*v_n = 0

becomes:

c_2*v_2 + ... + c_n*v_n = 0

so now, you are basically checking whether set of vectors:

v_2...v_n is linearly dependent.
0

Author Commented:
Thanks both.
0

Commented:
0 is a multiple of 5 and v_1 is a multiple of v_2, so they are  are linearly dependent
0
Question has a verified solution.

Are you are experiencing a similar issue? Get a personalized answer when you ask a related question.

Have a better answer? Share it in a comment.