A is a 2x2 matrices
and there is a transformation for which
there is a basis B={v,T(v)} for which
=
(0 1)
(-1 0)
prove that A is similar to
(0 1)
(-1 0)
matrices.
how i tried:
i got T being represented by base B
=
(0 1)
(-1 0)
i got T being represented by the stansdart base which is
A matrices.
what so A is similar to
(0 1)
(-1 0)
??
bacause of what?
is it correct?
this question is the third part of some bigger question in which i solved the first two parts.
V is lenear space dimV=n n>1
T:V->V is a transformation for wheach
i prooved that {T(v),v} basis
(0 1)
(-1 0)
in the third part i was asked:
A is 2x2 matrices
prove that A is similar to
(0 1)
(-1 0)
i am used to prove that if then they are similar
but here in the book they some thing liketwo representation in diffeerent basis
so they are similar
i cant understand this thing
why is that?
But the result is true for matrices with real entries. Let T be the linear transformation whose matrix (with respect to the standard basis) is A. Then , so T has no real eigenvalues and hence no (real) eigenvectors. It follows that if x is any nonzero vector then the vectors x and –Tx are linearly independent. If B is the basis consisting of those two vectors then the matrix of T with respect to B is So that matrix is similar to A.
Of course. These things happen when the formulation of the problem is not clear. For example in the answer #2 we already commented that for a basis satisfying the matrix of is (row representation) or equivalently, for is (column representation). Now in the third part of the problem it seems we have a matrix without a reference to the field .