Some Pieces of Linear Algebra(1)

 
1. oparation <=> multiply matrix
 
2. rank & information of vectors
 
we suppose every vector has its information.we use imformation content to discribe how much imformation a vector( or vectors or matrix…something like that) have.
for example :
arpha=[2 1] , we say arpha has 1 information content. we write it like this : IC(arpha)=1.
matrix A= [2 1,2 3] , we say A has 2 information content.(because A has two independent row vector)
 
thinks about this:
Does  my putting two vectors together will increase the information content they have?
for example:
arpha[2,1] beta=[1,1]
IC(arpha,beta)=2=Ic(arpha)+Ic(beta)
 
… of course, the answer is no.
does it will reduce the information content? …The answer is Yes!
look at this :
arpha[2,2] beta=[1,1]
IC(arpha,beta)=1<Ic(arpha)+Ic(beta)
 
the key here is Ic(arpha,beta)=?
so…Ic = What?
It’s a really simple quiz, Ic completely = rank!
of course , rank(arpha,beta)=1!
 
 
Remember it : rank means how much information the vectors have!
do you  know how conveniently this conclution uses?
 
for instance: 
 
operation could not increase the information a matrix have.
so, multiply matrix could not increase the rank of the matrix.
that means:
 R(AB)<=min{R(A),R(B)}
 
 
 
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