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Understanding Transvections:T_W:V-->V Tv=v+w
Esteemed Algebraists:
Please help me understand better the definition of a transvection.
Let V be a finite-dimensional vector space, and let W
be a codimension-1 subspace of V . A transvection
is defined to be an invertible linear map T:V-->V
such that:
i) T|_W =1_W , i.e., the restriction of T to W
is the identity on W.
ii)For any v in V, T(v)=v+w ; w in W.
Condition i) is clear, but does condition ii) just say that
vectors in V-W are mapped to V-W?
Also: given a choice of basis for V, is the
matrix representation for V always that of
a shear matrix, i.e., a matrix with all diagonal
entries equal to 1, and all off-diagonal entries
except for exactly one equal to zero, i.e., a
matrix describing adding a multiple by k of one
row to another row?
I know this is the representation in vector spaces over R; is
it true for V.Spaces over any field F? ( I know all V.Spaces of same
dimension are isomorphic, but I don't know if that guarantees the result).
I was thinking of a simple example of a linear map from
R<sup>3</sup> to R<sup>3</sup>
preserving points of types (x,0,0) and (0,y,0). Then ii) above would say that, using the
standard basis {e_<sub>i</sub>; i=1,2,3}.
i) T(1,0,0)=(1,0,0)
ii) T(0,1,0)=(0,1,0)
iii) T(0,0,1)= (0,0,1)+(a,b,0) ; a,b in F
Is the intended meaning that for z in V-W, T(z) in V-W? Also, the representation of
this transvection does not seem to match that of a shear transformation, since it includes
the case of two non-zero entries a,b.
Any Ideas?
Thanks in Advance.
Thanks.
Esteemed Algebraists:
Please help me understand better the definition of a transvection.
Let V be a finite-dimensional vector space, and let W
be a codimension-1 subspace of V . A transvection
is defined to be an invertible linear map T:V-->V
such that:
i) T|_W =1_W , i.e., the restriction of T to W
is the identity on W.
ii)For any v in V, T(v)=v+w ; w in W.
Condition i) is clear, but does condition ii) just say that
vectors in V-W are mapped to V-W?
Also: given a choice of basis for V, is the
matrix representation for V always that of
a shear matrix, i.e., a matrix with all diagonal
entries equal to 1, and all off-diagonal entries
except for exactly one equal to zero, i.e., a
matrix describing adding a multiple by k of one
row to another row?
I know this is the representation in vector spaces over R; is
it true for V.Spaces over any field F? ( I know all V.Spaces of same
dimension are isomorphic, but I don't know if that guarantees the result).
I was thinking of a simple example of a linear map from
R<sup>3</sup> to R<sup>3</sup>
preserving points of types (x,0,0) and (0,y,0). Then ii) above would say that, using the
standard basis {e_<sub>i</sub>; i=1,2,3}.
i) T(1,0,0)=(1,0,0)
ii) T(0,1,0)=(0,1,0)
iii) T(0,0,1)= (0,0,1)+(a,b,0) ; a,b in F
Is the intended meaning that for z in V-W, T(z) in V-W? Also, the representation of
this transvection does not seem to match that of a shear transformation, since it includes
the case of two non-zero entries a,b.
Any Ideas?
Thanks in Advance.
Thanks.
Last edited: