the rank one tensors’ basis changes
Tag Archives: linear algebra
Einstein-Penrose ‘s strong sum convention
Filed under math, mathematics, multilinear algebra, word algebra
wedge product example
When bivectors are defined by
,
so, for two generic covectors
and ,
we have the bivector
.
Otherwise,
Cf. this with the data and to construct the famous
So, nobody should be confused about the uses of the symbol dans le calcul vectoriel XD
change of basis and change of components
always it amuses me the incredulity of people when they receive an explanation of the covariant and contravariant behavior of linear algebra matters.
If one has a change of basis , you gotta to specify the basis that is assigned.
If you begin by chosing the canonical basis , where
, ,
and another basis, say
, ,
then we have and . From here one can resolve: and .
So if a vector is an arbitrary (think… ) element in then its expression in the new basis is:
.
Remember .
So, if are the component in the old basis, then are the components in the new one.
Matricially what we see is this:
.
This corresponds to the relation
which -contrasted against in the change between those basis- causes some mind twist :D… :p
One says then that the base vectors co-vary, but components of vectors contra-vary.
For more: Covariance and contravariance of vectors.
Filed under algebra, multilinear algebra
simu exam comment…
The inversion of a matrix.
For a change of basis in :
we have as a change-of-basis-matrix:
and by solving for:
then we get as an inverse of that matrix:
All that inside a recent exam did at the dept of maths
Filed under algebra, mathematics, multilinear algebra, what is mathematics
real quadratic forms
A quadratic form in an -dimensional real vectorspace , is a bilineal map which can be determined by a -matrix via
If we are allowed to write we can find that he (or she) possibly satisfy
-
, property dubbed symmetry
-
, positive-definiteness
-
iff , non-degeneracy
In case of affirmatively both three are satisfy is called a metric on and the pair
is called (real) Euclidean vectorspace…
MORE on
MORE on
Filed under math, multilinear algebra
matrices especiales de 2×2, módulo 2 y módulo 3
no es difícil calcular que tiene seis elementos y que este grupo es el grupo simétrico .
¿Y que hay acerca de ? también no es difícil calcular .
¿Es cierto qué ?
Filed under algebra, group theory