Tag Archives: linear algebra

Einstein-Penrose ‘s strong sum convention


the rank one tensors’ basis changes

abstractrelativity017

Advertisements

2 Comments

Filed under math, mathematics, multilinear algebra, word algebra

wedge product example


When bivectors are defined by

\beta^i\wedge\beta^j:=\beta^i\otimes\beta^j-\beta^j\otimes\beta^i,

so, for two generic covectors

\theta=a\beta^1+b\beta^2+c\beta^3 and \phi=d\beta^1+e\beta^2+f\beta^3,

we have the bivector

\theta\wedge\phi=(bf-ce)\beta^2\wedge\beta^3+(cd-af)\beta^3\wedge\beta^1+(ad-be)\beta^1\wedge\beta^2.

 

Otherwise,

Cf. this with the data \left(\begin{array}{c}a\\b\\c\end{array}\right) and \left(\begin{array}{c}d\\e\\f\end{array}\right) to construct the famous

\left(\begin{array}{c}a\\b\\c\end{array}\right)\times\left(\begin{array}{c}d\\e\\f\end{array}\right)=\left(\begin{array}{c}bf-ce\\cd-af\\ad-be\end{array}\right)

So, nobody should be confused about the uses of the symbol \wedge dans le calcul vectoriel XD

2 Comments

Filed under algebra, cucei math, differential geometry, math analysis, mathematics, multilinear algebra, what is math, word algebra

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 \mathbb{R}^2\to\mathbb{R}^2, you gotta to specify the basis that is assigned.

If you begin by chosing the canonical basis e_1,e_2, where

e_1=\left(\!\!\begin{array}{c}1\\ 0\end{array}\!\!\right) , e_2=\left(\!\!\begin{array}{c}0\\ 1\end{array}\!\!\right),

and another basis, say

b_1=\left(\!\!\begin{array}{c}1\\ 0\end{array}\!\!\right) , b_2=\left(\!\!\begin{array}{c}1\\ 1\end{array}\!\!\right),

then we have b_1=e_1 and b_2=e_1+e_2. From here one can resolve: e_1=b_1 and e_2=-b_1+b_2.

So if a vector v=Ae_1+Be_2 is an arbitrary (think… A,B\in\mathbb{R}) element in \mathbb{R}^2 then its expression in the new basis is:

v=Ab_1+B(-b_1+b_2)

=(A-B)b_1+Bb_2.

Remember A,B\in\mathbb{R}.

So, if v=\left(\!\!\begin{array}{c}A\\ B\end{array}\!\!\right)_e are the component in the old basis, then v=\left(\!\!\begin{array}{c}A-B\\ B\end{array}\!\!\right)_b are the components in the new one.

Matricially what we see is this:

\left(\!\!\begin{array}{cc}1&-1\\ 0&1\end{array}\!\!\right)\left(\!\!\begin{array}{c}A\\ B\end{array}\!\!\right)=\left(\!\!\begin{array}{c}A-B\\ B\end{array}\!\!\right).

This corresponds to the relation

M^{-1}v_e=v_b, \qquad (*)

which -contrasted against Me_i=b_i 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.

1 Comment

Filed under algebra, multilinear algebra

simu exam comment…


The inversion of a matrix.

For a change of basis in \mathbb{R}^3:

\eta_1=\varepsilon_1

\eta_2=\varepsilon_1+2\varepsilon_2

\eta_3=\varepsilon_2-\varepsilon_3

we have as a change-of-basis-matrix:

\left(\begin{array}{ccc}1&1&0\\ 0&2&1\\ 0&0&-1\end{array}\right)

and by solving for:

\varepsilon_1=\eta_1

\varepsilon_2=-\frac{1}{2}\eta_1+\frac{1}{2}\eta_2

\varepsilon_2=-\frac{1}{2}\eta_1+\frac{1}{2}\eta_2-\eta_3

then we get as an inverse of that matrix:

\left(\begin{array}{ccc}1&-1/2&-1/2\\ 0&1/2&1/2\\ 0&0&-1\end{array}\right)

All that inside a recent exam did at the dept of maths

Leave a comment

Filed under algebra, mathematics, multilinear algebra, what is mathematics

real quadratic forms


A quadratic form in an m -dimensional  real vectorspace V, is a bilineal map V\times V\to\mathbb{R} which can be determined by a m\times m-matrix A via

(v,w)\mapsto v^TAw

If we are allowed to write g(v,w)=v^TAw we can find that he (or she) possibly satisfy

  1. g(v,w)=g(w,v),  property dubbed symmetry
  2. g(v,v)\ge0, positive-definiteness
  3. g(v,v)=0 iff v=0, non-degeneracy

In case of affirmatively both three are satisfy g is called a metric on V and the pair

(V,g)

is called (real) Euclidean vectorspace…

MORE on \mathbb{R}^2

MORE on \mathbb{R}^3

2 Comments

Filed under math, multilinear algebra

a GPS for a surface to do calculus on it


 

1 Comment

Filed under math

matrices especiales de 2×2, módulo 2 y módulo 3


no es difícil calcular que SL_2(\mathbb{Z}_2) tiene seis elementos y que este grupo es el grupo simétrico S_3=\langle a,b\mid a^2=b^3=e,\ ab^2=ba\rangle.

¿Y que hay acerca de SL_2(\mathbb{Z}_3)? también no es difícil calcular |SL_2(\mathbb{Z}_3)|=24.

¿Es cierto qué SL_2(\mathbb{Z}_3)=S_6?

Leave a comment

Filed under algebra, group theory