Let β and β ′ be bases for the finite dimensional vector space V of dimension n over the field F, and let Q = [ I V ] β ′ β , where I V is the identity operator on V. I just recently proved that Q [ x ] β ′ = [ x ] β for every x ∈ V (twas rather simple), which suggests the title of "basis transformation" matrix or "coordinate transformation" matrix for the matrix Q. I am now wondering whether the converse holds.Let V ′ denote V with its elements written with respect to the basis β ′ , and suppose that Q ∈ M n ( F ) is such that Q [ x ] β ′ = [ x ] β for every x. Since L ( V ′ V ) is isomorphic to the matrix algebra M n ( F ) by sending a linear operator to its matrix representation, given Q there exists a linear operator T ∈ L ( V ′ , V ) such that Q = [ T ] β ′ β . Hence [ T ] β ′ β [ x ] β ′ = [ x ] β or [ T ( x ) ] β = [ x ] β ...I want to say that this implies T = I V , but I can't clearly see what lemma I need in order to make that conclusion.

Question

Let   β and       β    ′   be bases for the finite dimensional vector space   V of dimension n over the field   F, and let   Q  =  [      I    V        ]            β      ′          β  , where       I    V   is the identity operator on   V. I just recently proved that   Q  [  x      ]            β      ′        =  [  x      ]      β   for every   x  ∈  V (twas rather simple), which suggests the title of &quot;basis transformation&quot; matrix or &quot;coordinate transformation&quot; matrix for the matrix   Q. I am now wondering whether the converse holds.Let       V    ′   denote   V with its elements written with respect to the basis       β    ′  , and suppose that   Q  ∈      M    n    (  F  ) is such that   Q  [  x      ]            β      ′        =  [  x      ]      β   for every   x. Since   L  (      V    ′    V  ) is isomorphic to the matrix algebra       M    n    (  F  ) by sending a linear operator to its matrix representation, given   Q there exists a linear operator   T  ∈  L  (      V    ′    ,  V  ) such that   Q  =  [  T      ]            β      ′          β  . Hence   [  T      ]            β      ′          β    [  x      ]            β      ′        =  [  x      ]      β   or   [  T  (  x  )      ]      β    =  [  x      ]      β  ...I want to say that this implies   T  =      I    V  , but I can&#039;t clearly see what lemma I need in order to make that conclusion.

Kaya Kemp · Accepted Answer

Your equation   [  T  (  x  )      ]    β    =  [  x      ]    β   says that   T  (  x  ) and   x have the same coordinates in the basis   β. This means they can be written as the same linear combination of vectors of   β and thus   T  (  x  )  =  x for all   x  ∈  V, i.e.   T  =      I    V  .

Let β and β ′ </msup> be bases for

Answered question

Answer & Explanation

New Questions in Linear algebra