KungFu
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- TL;DR
- I want to find the isometric basis corresponding to the musical isomorphism between the vector space E = ##R^3## and its dual space ##E^* = (R^3)^*) ## .
Im very new to the terminologies of isometric basis and musical isomorphism, will appreciate a lot if someone could explain this for me in a simple way for a guy with limited experience in this field.
The concrete problem I want to figure out is this:
Given:
Let ##v_1 = (1,0,0) , v_2 = (1,1,0), v_3 = (0,1,1)## be a basis in the vector space ##E = R^3##
problem:
Find a dual basis to ##{v_1,v_2,v_3} ## in ##E^*## = ##(R^3)^*## and the isometric basis corresponding to the musical isomorphism between E and E*.
I have found the dual basis: I did this by using the property ##(v^*)^i(v_j) = \delta_{ij}##, where ##(v^*)^i## is the ith covector in the dual basis.
I represent the dual basis as a linear combination of the orthonormal basis in ##(R^3)^*##, the dual basis is then
##(v_1)^* = (1,-1,1)##, ##(v_2)^* = (0,1,-1)## and ##(v_3)^* = (0,0,1) ##
now, to my question : how is the isometric basis corresponding to the musical isomorphism between E and ##E^*## defined ?
and even more basic : what do we mean be an isometric basis, and what do we mean by a musical isomorphism, you can as well try to explain what an isomorphism is first, in simple words ;)
The concrete problem I want to figure out is this:
Given:
Let ##v_1 = (1,0,0) , v_2 = (1,1,0), v_3 = (0,1,1)## be a basis in the vector space ##E = R^3##
problem:
Find a dual basis to ##{v_1,v_2,v_3} ## in ##E^*## = ##(R^3)^*## and the isometric basis corresponding to the musical isomorphism between E and E*.
I have found the dual basis: I did this by using the property ##(v^*)^i(v_j) = \delta_{ij}##, where ##(v^*)^i## is the ith covector in the dual basis.
I represent the dual basis as a linear combination of the orthonormal basis in ##(R^3)^*##, the dual basis is then
##(v_1)^* = (1,-1,1)##, ##(v_2)^* = (0,1,-1)## and ##(v_3)^* = (0,0,1) ##
now, to my question : how is the isometric basis corresponding to the musical isomorphism between E and ##E^*## defined ?
and even more basic : what do we mean be an isometric basis, and what do we mean by a musical isomorphism, you can as well try to explain what an isomorphism is first, in simple words ;)