A = ( k − 2 1 − k k ) , where k is a constant A transformation T : R 2 → R 2 is represented by the matrix A. Find the value of k for which the line y = 2 x is mapped onto itself under T.

Question

A  =      (                            k                          −          2                                      1          −          k                          k                      )    , where k is a constant  A transformation   T  :        R    2    →        R    2     is  represented by the matrix A.  Find the value of k for which the line   y  =  2  x   is mapped onto itself under T.

Valeria Wolfe · Accepted Answer

The line   y  =  2  x is mapped to itself if we have that for each point   (  x  ,  y  ) on the line   T  (  x  ,  y  ) also lies on the line. We needn&#039;t have   T  (  x  ,  2  x  )  =  (  x  ,  2  x  ) for all   x, it suffices to have   T  (  x  ,  2  x  )  =        (    ϕ  (  x  )  ,  2  ϕ  (  x  )        )   for all   x and some function   ϕ.Having this in mind, we look at   T  (  x  ,  2  x  )  =        (    x  (  k  −  4  )  ,  x  (  k  +  1  )        )  . We must have   2  (  k  −  4  )  =  k  +  1 for   (  x  (  k  −  1  )  ,  x  (  k  +  1  )  ) to lie on our line. This gives   k  =  9

racodelitusmn · Accepted Answer

On the same line (!) of thought: the line     l  :  y  =  2  x   is the same as the vector space     Span  ⁡  {  (  1  ,  2  )  }  ≤        R    2   , or if you prefer:     l  :  {  (  r  ,  2  r  )    /    r  ∈  R  }  , and then what we really want to happen is      (                            k                          −          2                                      1          −          k                          k                      )        (                            1                                      2                      )    =      (                            r                                      2          r                      )    ⟹                    k        =                                r        +        4                            k        =        2        r        −        1            so     r  +  4  =  2  r  −  1  ⟹  r  =  5   and thus     k  =  9

A = ( <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd>

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