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I am reading Segei Winitzki's book: Linear Algebra via Exterior Products ...
I am currently focused on Section 1.6: Dual (conjugate) vector space ... ...
I need help in order to get a clear understanding of the notion or concept of coefficients of a vector [itex]v[/itex] as linear functions (covectors) of the vector [itex]v[/itex] ...
The relevant part of Winitzki's text reads as follows:
In the above text we read:
" ... ... So the coefficients [itex]v_k, \ 1 \leq k \leq n[/itex], are linear functions of the vector [itex]v[/itex] ; therefore they are covectors ... ... "
Now, how and in what way exactly are the coefficients [itex]v_k[/itex] a function of the vector [itex]v[/itex] ... ... ?
To indicate my confusion ... if the coefficient [itex]v_k[/itex] is a linear function of the vector [itex]v[/itex] then [itex]v_k(v)[/itex] must be equal to something ... but what? ... indeed what does [itex]v_k(v)[/itex] mean? ... further, what, if anything, would [itex]v_k(w)[/itex] mean where [itex]w[/itex] is any other vector? ... and further yet, how do we formally and rigorously prove that [itex]v_k[/itex] is linear? ... what would the formal proof look like?... ...
Hope someone can help ...
Peter
============================================================================
*** NOTE ***
To indicate Winitzki's approach to the dual space and his notation I am providing the text of his introduction to Section 1.6 on the dual or conjugate vector space ... ... as follows ... ...
I am currently focused on Section 1.6: Dual (conjugate) vector space ... ...
I need help in order to get a clear understanding of the notion or concept of coefficients of a vector [itex]v[/itex] as linear functions (covectors) of the vector [itex]v[/itex] ...
The relevant part of Winitzki's text reads as follows:
In the above text we read:
" ... ... So the coefficients [itex]v_k, \ 1 \leq k \leq n[/itex], are linear functions of the vector [itex]v[/itex] ; therefore they are covectors ... ... "
Now, how and in what way exactly are the coefficients [itex]v_k[/itex] a function of the vector [itex]v[/itex] ... ... ?
To indicate my confusion ... if the coefficient [itex]v_k[/itex] is a linear function of the vector [itex]v[/itex] then [itex]v_k(v)[/itex] must be equal to something ... but what? ... indeed what does [itex]v_k(v)[/itex] mean? ... further, what, if anything, would [itex]v_k(w)[/itex] mean where [itex]w[/itex] is any other vector? ... and further yet, how do we formally and rigorously prove that [itex]v_k[/itex] is linear? ... what would the formal proof look like?... ...
Hope someone can help ...
Peter
============================================================================
*** NOTE ***
To indicate Winitzki's approach to the dual space and his notation I am providing the text of his introduction to Section 1.6 on the dual or conjugate vector space ... ... as follows ... ...
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