Say you have v → 1 = [ 1 2 3 ] and v → 3 = anything.Is the set { v → 1 , v → 2 , v → 3 } always considered linearly dependent because { v → 1 , v → 2 } is?

Question

Say you have                     v        →              1    =      [                            1                                      2                                      3                      ]   and                     v        →              3    = anything.Is the set   {                    v        →              1    ,                    v        →              2    ,                    v        →              3    } always considered linearly dependent because   {                    v        →              1    ,                    v        →              2    } is?

agergadas3b · Accepted Answer

The requirement for linear dependence is that       a    1        v    1    +      a    2        v    2    +      a    3        v    3    =  0, where not all the as are zero. So if       a    1    =      a    2   and       a    3    =  0, then that would make it zero, so yes, they will always be linearly dependent

Brooklyn Farrell · Accepted Answer

If you want to check by the definition:  1  ⋅                    v        1            →        +  1  ⋅                    v        2            →        +  0  ⋅                    v        3            →        =            0      →      and not all the coefficients are zero, thus there is lin. dependence.

Say you have vec(v)_1=[[1],[2],[3]] \ vec(v)_2=[[-1],[-2],[-3]] and vec(v)_3= anything. Is the set {vec(v)_1,vec(v)_2,vec(v)_3} always considered linearly dependent because {vec(v)_1,vec(v)_2} is?

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