representative matrix of a linear transformationLet T : F 2 → F 2 a linear tranformation such that T 2 = 0, Prove that T = 0, or there exists a basis of M 2 ( F ) in which the representative matrix is: [ 0 1 0 0 ] .

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representative matrix of a linear transformationLet   T  :        F    2    →        F    2   a linear tranformation such that       T    2    =  0, Prove that   T  =  0, or there exists a basis of       M    2    (  F  ) in which the representative matrix is:       [                            0                          1                                      0                          0                      ]  .

stellak012s7aoc · Accepted Answer

Assume that   T  ≠  0 then there&#039;s   u  ∈        F    2   such that   T  (  u  )  =  v  ≠  0. Prove that   B  =  (  v  ,  u  ) is a basis of         F    2   and the matrix of   T in this basis has the desired form.

conIjonnoraj0nls · Accepted Answer

We have  σ  (  σ  (  A  )  )  =  σ  (  2  A  −  3      A    T    )  =  2  (  2  A  −  3      A    T    )  −  3  (  2      A    T    −  3  A  )    =  13  A  −  12      A    T    =  13  A  +  4  (  σ  (  A  )  −  2  A  )  =  5  A  +  4  σ  (  A  )hence the polynomial       x    2    −  4  x  −  5  =  (  x  +  1  )  (  x  −  5  ) with simple roots −1 and 5 annihilates   σ so   σ is diagonalized and since   σ  ≠  k      i    d   hence −1 and 5 are two eigenvalues of   σ. Now to find the multiplicity of the both eigenvalues we solve the equality  σ  (  A  )  =  k  A    k  =  −  1  ,  5for example for   k  =  −  1 we find  2  A  −  3      A    T    =  −  A    ⟺    A  =      A    T      ⟺    A  ∈      S    n  and we know that  dim  ⁡      S    n    =            n      (      n      +      1      )        2  and we find also that the multiplicity of 5 is   dim  ⁡      A    n    =            n      (      n      −      1      )        2   and we conclude.

representative matrix of a linear transformation Let T : <mi mathvariant="double-str

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