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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
0
InteractiveMind
Asked:
InteractiveMind
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4 Solutions
 
ozoCommented:
v_1=(0,0,0) is not independent of v_2, v_3
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UrosVidojevicCommented:
>> 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.
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ozoCommented:
Determinant of
| 0 v21 v31 |
| 0 v22 v32 |
| 0 v23 v33 |
is 0
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InteractiveMindAuthor Commented:
Thanks both.
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ozoCommented:
0 is a multiple of 5 and v_1 is a multiple of v_2, so they are  are linearly dependent
0

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