Determine the minimal polynomial of T with respect to ( 0 0 1 ) t ..Let T : R 3 → R 3 be defined by T ( v ) = ( 2 − 1 1 − 3 4 − 5 − 3 3 − 4 ) v .Determine the minimal polynomial of T with respect to ( 0 0 1 ) . T ( v ) = ( 2 − 1 1 − 3 4 − 5 − 3 3 − 4 ) ( 0 0 1 ) = ( 1 − 5 − 4 ) T 2 ( v ) = ( 2 − 1 1 − 3 4 − 5 − 3 3 − 4 ) ( 1 − 5 − 4 ) = ( 3 − 3 − 2 ) T 3 ( v ) = ( 2 − 1 1 − 3 4 − 5 − 3 3 − 4 ) ( 3 − 3 − 2 ) = ( 7 − 11 − 10 ) The corresponding matrix is A = ( v , T ( v ) , T 2 ( v ) , T 3 ( v ) ) = ( 0 1 3 7 0 − 5 − 3 − 11 1 − 4 − 2 − 10 ) After row reducing this matrix, we obtain ( 1 0 0 − 2 0 1 0 1 0 0 1 2 ) , and so null ( A ) = span { ( 0 0 1 0 ) , ( 2 − 1 0 1 ) } . From this we obtain polynomials f ( x ) = x 2 and g ( x ) = x 3 − x − 2 and since deg ⁡ f ( x ) &lt; deg ⁡ g ( x ) ,, x 2 is the minimal polynomial.The back of my book says the minimal polynomial is x 2 − 3 x + 2. What am I doing wrong?

Question

Determine the minimal polynomial of T with respect to             (                                    0                                0                                1                              )        t    ..Let   T  :            R        3    →            R        3   be defined by   T  (  v  )  =      (                            2                          −          1                          1                                      −          3                          4                          −          5                                      −          3                          3                          −          4                      )    v  .Determine the minimal polynomial of T with respect to       (                            0                                      0                                      1                      )    .  T  (  v  )  =      (                            2                          −          1                          1                                      −          3                          4                          −          5                                      −          3                          3                          −          4                      )        (                            0                                      0                                      1                      )    =      (                            1                                      −          5                                      −          4                      )        T    2    (  v  )  =      (                            2                          −          1                          1                                      −          3                          4                          −          5                                      −          3                          3                          −          4                      )        (                            1                                      −          5                                      −          4                      )    =      (                            3                                      −          3                                      −          2                      )        T    3    (  v  )  =      (                            2                          −          1                          1                                      −          3                          4                          −          5                                      −          3                          3                          −          4                      )        (                            3                                      −          3                                      −          2                      )    =      (                            7                                      −          11                                      −          10                      )  The corresponding matrix is  A  =  (  v  ,  T  (  v  )  ,      T    2    (  v  )  ,      T    3    (  v  )  )  =      (                            0                          1                          3                          7                                      0                          −          5                          −          3                          −          11                                      1                          −          4                          −          2                          −          10                      )  After row reducing this matrix, we obtain       (                            1                          0                          0                          −          2                                      0                          1                          0                          1                                      0                          0                          1                          2                      )  , and so   null  (  A  )  =  span      {          (                                    0                                                0                                                1                                                0                              )        ,          (                                    2                                                −            1                                                0                                                1                              )        }    . From this we obtain polynomials   f  (  x  )  =      x    2   and   g  (  x  )  =      x    3    −  x  −  2 and since   deg  ⁡  f  (  x  )  &amp;lt;  deg  ⁡  g  (  x  )  ,,       x    2   is the minimal polynomial.The back of my book says the minimal polynomial is       x    2    −  3  x  +  2. What am I doing wrong?

Izabella Henson · Accepted Answer

Step 1You got the nullspace of A wrong. It is one-dimensional, since the rank of A is 3. The row-reduced form that you have indicates that the first three columns are linearly independent, and the fourth one is a certain linear combination of them. This yields  null  ⁡  (  A  )  =  span  ⁡  (  (  2  ,  −  1  ,  −  2  ,  1      )    t    )and the minimal polynomial of v is  p  (  x  )  =      x    3    −  2      x    2    −  x  +  2Step 2Being of the same degree as the order of the matrix, this is also its minimal polynomial, and characteristic polynomial.I don&#039;t know where the book&#039;s answer comes from: it&#039;s easy to check that       T    2    v  −  3  T  v  +  2  v  ≠  0.

Determine the minimal polynomial of T with respect to (001)^t.

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