Answer to Question #215531 in Abstract Algebra for Sujata

Question

If V is a finite dimensional vector space over F and T and S are linear transformation on V, then there exist two ordered bases A and B for V such that A =B,where A is matrix of T and B is matrix of S,then prove that A and B are similar. Deduce that det(A) = det(B).

Answer 1

Solution analysis;

If we deduce that det(A)=det(B)

It means that transformation T and S stretch vector space V represented by the vector base in the same amount;

S,T:V₁(F)$\to$ V₂(F)

If A and B are ordered basis of V ,

Let a matrix C be the change from A to B,then;

det ([V_B])=det C^-1det[V_A]detC=det([V_A])

This means that transformation T and S are similar.

Therefore,the matrix of transformation should also be similar;

B=C^-1AC

Since C^-1C=1

B=A