Are two 2D vectors linearly dependent when x 1 y 2 = x 2 y 1 ?I was working on a proof for why a set of three vectors ∈ R 2 are always linearly dependant when I came along this.It appears that given two vectors [ x 1 , x 2 ] T and [ y 1 , y 2 ] T , if x 1 y 2 = x 2 y 1 then one vector is a scalar multiple of the other. This is the case in my proof where you have to divide by zero (more accurately, divide by x 1 y 2 − x 2 y 1 ) so I feel like this can't be a coincidence.An example is when x 1 y 2 = x 2 y 1 = 12, we could make pairs of vectors like v 1 = [ 12 , 4 ] T , v 2 = [ 3 , 1 ] T and then v 1 = 4 v 2 Does this hold for all vector pairs that satisfy the restriction and if so, why?

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Are two 2D vectors linearly dependent when       x    1        y    2    =      x    2        y    1  ?I was working on a proof for why a set of three vectors   ∈            R        2   are always linearly dependant when I came along this.It appears that given two vectors   [      x    1    ,      x    2        ]    T   and   [      y    1    ,      y    2        ]    T  , if       x    1        y    2    =      x    2        y    1   then one vector is a scalar multiple of the other. This is the case in my proof where you have to divide by zero (more accurately, divide by       x    1        y    2    −      x    2        y    1  ) so I feel like this can&#039;t be a coincidence.An example is when       x    1        y    2    =      x    2        y    1    =  12, we could make pairs of vectors like       v    1    =  [  12  ,  4      ]    T    ,      v    2    =  [  3  ,  1      ]    T   and then       v    1    =  4      v    2  Does this hold for all vector pairs that satisfy the restriction and if so, why?

vibrerentb · Accepted Answer

Yes. Just stack the vectors on a matrix M and notice that   d  e  t  (  M  )  =      x    1        y    2    −      x    2        y    1  . So, if       x    1        y    2    =      x    2        y    1   we have   d  e  t  (  M  )  =  0 and then the vectors are linearly dependent.

pominjaneh6 · Accepted Answer

Given       x    1        y    2    =      x    2        y    1  . If there exists a zero component, then the set of that two vectors must be linearly dependentAssume that no components are zero. So            y      2              y      1        =            x      2              x      1        =  kfor some scalar k.From above equation, we get       y    2    =  k      y    1   and       x    2    =  k      x    1  . Note that                    (                  y          1                ,                  y          2                )                            =        (                  y          1                ,        k                  y          1                )                                          =                  y          1                (        1        ,        k        )                                          =                              y            1                                x            1                                    x          1                (        1        ,        k        )                                          =                              y            1                                x            1                          (                  x          1                ,        k                  x          1                )                                          =                              y            1                                x            1                          (                  x          1                ,                  x          2                )            Hence, the set   {  (      y    1    ,      y    2    )  ,  (      x    1    ,      x    2    )  } is linearly dependent.

Are two 2D vectors linearly dependent when x_1y_2=x_2y_1?

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