From my understanding, this question asks me is there any formula which can make T ( x ) = x 2 a non-linear transformation into a matrix transformation ? What am I supposed to answer to this question?

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From my understanding, this question asks me is there any formula which can make   T  (  x  )  =      x    2   a non-linear transformation into a matrix transformation ? What am I supposed to answer to this question?

sag2y8s · Accepted Answer

I think that the question is rather poor. There is no linear transformation S such that   S  (  x  )  =      x    2  , so there is no linear transformation that agrees with T everywhere.On the other hand, there is a &quot;natural&quot; way to obtain a linear transformation S from   T  (  x  )  =      x    2  , and that would be to take the derivative. The derivative   S  (  x  )  =      T    ′    (  x  )  =  2  x is the linear map which scales vectors by More generally, given a &quot;non-linear&quot; (but still differentiable) map   T  :        R    n    →        R    m  , written in the standard bases as   T  (      x    1    ,  …  ,      x    n    )  =        (        T    1    (      x    1    ,  …  ,      x    n    )  ,  …  ,      T    m    (      x    1    ,  …  ,      x    n    )        )    ,the derivative   D      T    p   at any point   p  =  (      p    1    ,  …  ,      p    n    ) is a linear map         R    n    →        R    m  .  Moreover, in the standard bases,   D      T    p   can be identified with the   m  ×  n Jacobian matrix of partial derivatives:  D      T    p    =      (                                                              ∂                              T                1                                                    ∂                              x                1                                              (          p          )                                                            ∂                              T                1                                                    ∂                              x                2                                              (          p          )                          …                                                            ∂                              T                1                                                    ∂                              x                n                                              (          p          )                                                                        ∂                              T                2                                                    ∂                              x                1                                              (          p          )                                                            ∂                              T                2                                                    ∂                              x                2                                              (          p          )                          …                                                            ∂                              T                2                                                    ∂                              x                n                                              (          p          )                                      ⋮                          ⋮                          ⋱                          ⋮                                                                        ∂                              T                m                                                    ∂                              x                1                                              (          p          )                                                            ∂                              T                m                                                    ∂                              x                2                                              (          p          )                          …                                                            ∂                              T                m                                                    ∂                              x                n                                              (          p          )                      )    .The upshot is that as long as we look at points   q  =  (      q    1    ,  …  ,      q    n    ) that are &quot;very small,&quot; then the affine transformation   q  ↦  T  (  p  )  +  D      T    p    (  q  ) closely approximates the (in general nonlinear) map   q  ↦  T  (  p  +  q  )So, for the example   T  :  R  →  R defined by   T  (  x  )  =      x    2  , we get a linear map at every point p defined as   D      T    p   which we typically just write as       T    ′    (  p  ) which is really just a   1  ×  1 matrix       T    ′    (  p  )  =  (  2  p  ) . As a linear map acting on   1  ×  1 column vectors (q), we have  D      T    p    (  q  )  =      T    ′    (  p  )  (  q  )  =  (  2  p  )  (  q  )  =  (  2  p  q  )  .Hence the end result is that &quot;vectors&quot; (q) are mapped linearly to (2pq) by the derivative, and if we identify   1  ×  1 matrices with numbers,                    T        (        p        +        q        )        =        (        p        +        q                  )          2                                    =                  p          2                +        2        p        q        +                  q          2                                                  ≈                  p          2                +        2        p        q                                  (small numbers like                   q           are &quot;negligible&quot; if we square them)                                                  =        T        (        p        )        +        2        p        q                                          =        T        (        p        )        +                  T          ′                (        p        )        q        ,            so long as q is sufficiently small. You may recognize that   T  (  p  )  +      T    ′    (  p  )  q is really the first-order Taylor polynomial of T at p

From my understanding, this question asks me is there any formula which can make T ( x

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