Question

How can you tell whether vectors are linear independent? such as v_1 = (0, 0, 2, 2) v_2=(3, 3, 0, 0) v_3=(1,1,0,-1) v_4=(4, 4, 2, 1)

Answer 1

A linearly DEPENDENT vector is a scalar multiple of another. So for example, consider:
\[v=<1,1,1>\]
If you had the vector <3,3,3> it would be:
\[<3,3,3>=<3(1),3(1),3(1)>=3v\]
So the vectors are linearly dependent.
However, the vector <3,3,2> is linearly independent.

Answer 2

A set of vectors \[B=\{v_1, v_2, v_3, ..., v_n\}\] if for \[a_1v_1 + a_2v_2 + ... + a_nv_n = 0\] only when \[a_i = 0\] for all 1 < i < n

In other words no vector in the set can be expressed as a linear combination of the rest of the vectors.

Answer 3

so the only way it can be linearly independent if you can't multiply any scalar to get the same vector

Answer 4

so in my case it would be linearly independent?

Answer 5

no

Answer 6

How come? You can't multiply all the vectors by a scalar to get all the same vector

Answer 7

(0, 0, 2, 2) +(3, 3, 0, 0) +(1,1,0,-1) -(4, 4, 2, 1)=(0,0,0,0)

Answer 8

Looks dependent to me

Answer 9

slightly confused now so you're saying if we add them and get the zero vector it's dependent?

Answer 10

How about in the case of V=<1,1> and v_1=<2,2>? Aren't they dependent but when added it's not a zero vector

Answer 11

a nontrivial linear combination yields the zero vector...thus they are dependent

Answer 12

2*(1,1)+(-1)(2,2)=(0,0)

Answer 13

okay I think I get it now thanks!

Answer 14

Mine is somewhat of a specific case. Using the combination of <1,1,1>, <1,1,1>, and <1,1,1> you can get <3,3,3> that I have above. Or treat it as a scalar multiple.