How can I solve the following exercise?Let V be the vector space defined on the field R of n −dimensional. Let T : V → V be a linear transformation such that T n − 1 ≠ 0, T n = 0 and B = { x , T ( x ) , T 2 ( x ) , ⋯ , T n − 1 ( x ) } it is a basis for V. Find a matrix associated with the linear transformation T.I think the only way to explicitly determine the matrix associated with the linear transformation T, is to define T as T ( α 1 x + α 2 T ( x ) + ⋯ + α n − 1 T n − 1 ( x ) ) = α 1 x + α 2 T ( x ) + ⋯ + α n − 1 T n − 1 ( x ) .

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How can I solve the following exercise?Let   V be the vector space defined on the field   R of   n  −dimensional. Let   T  :  V  →  V be a linear transformation such that       T          n      −      1        ≠  0,       T      n    =  0 and   B  =      {    x    ,    T          (      x      )        ,          T          2              (      x      )        ,    ⋯    ,          T              n        −        1                    (      x      )        }   it is a basis for   V. Find a matrix associated with the linear transformation   T.I think the only way to explicitly determine the matrix associated with the linear transformation   T, is to define   T as   T      (          α          1        x    +          α          2        T          (      x      )        +    ⋯    +          α              n        −        1                    T              n        −        1                    (      x      )        )    =      α      1    x  +      α      2    T      (    x    )    +  ⋯  +      α          n      −      1            T          n      −      1            (    x    )  .

Annabella Velez · Accepted Answer

We have  T  [      α    1    x  +      α    2    T  (  x  )  +  ⋯  +      α    n        T          n      −      1        (  x  )  ]  =      α      1    T  (  x  )  +      α    2        T    2    (  x  )  +  ⋯  +      α          n      −      1            T          n      −      1        (  x  )To put it another way: if we rewrite   B as   B  =  {      v    1    ,      v    2    ,  …  ,      v    n    }, then we have  T  [      α    1        v    1    +      α    2        v    2    +  ⋯  +      α    n        v    n    ]  =      α      1        v    2    +      α    2        v    3    +  ⋯  +      α          n      −      1            v          n      −      1      Thus, the matrix of   T with respect to the basis   V is      (                            0                          0                          0                          ⋯                          0                                      1                          0                          0                          ⋱                          ⋮                                      0                          ⋱                          ⋱                          ⋱                          0                                      ⋮                          ⋱                          1                          0                          0                                      0                          ⋯                          0                          1                          0                      )

How can I solve the following exercise? Let V be the vector space defined on the field <

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