Math Amateur
Gold Member
MHB
- 3,920
- 48
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 an aspect of the notion or concept of the dual basis $$ \{ e^*_1, e^*_2, \ ... \ ... \ , \ e^*_n \} $$
The relevant part of Winitzki's text reads as follows:http://mathhelpboards.com/attachments/linear-abstract-algebra-14/5348-dual-vector-space-dual-basis-another-question-winitzki-section-1-6-a-winitzki-dual-basis-pngIn the above quoted text from Winitzki, we read:
" ... ... Please note that $$e^*_1$$ depends on the entire basis $$\{ e^*_j \}$$ and not only on $$e^*_1$$, as might appear from the notation $$e^*_1$$. ... ... "I am puzzled by this statement ... can someone explain how, and indeed why, $$e^*_1$$ depends on the entire basis $$\{ e^*_j \}$$ and not only on $$e^*_1$$ ... a clarification of the nature of the dual basis would be most helpful ...
Hope someone can help ...
Peter===========================================================*** NOTE ***To indicate Winitzki's approach to the dual space (and its basis) and his notation I am providing the text of his introduction to Section 1.6 on the dual or conjugate vector space ... ... as follows ... ...
http://mathhelpboards.com/attachments/linear-abstract-algebra-14/5349-dual-vector-space-dual-basis-another-question-winitzki-section-1-6-a-winitzki-1-section-1-6-part-1-png
http://mathhelpboards.com/attachments/linear-abstract-algebra-14/5350-dual-vector-space-dual-basis-another-question-winitzki-section-1-6-a-winitzki-2-section-1-6-part-2-png
I am currently focused on Section 1.6: Dual (conjugate) vector space ... ...
I need help in order to get a clear understanding of an aspect of the notion or concept of the dual basis $$ \{ e^*_1, e^*_2, \ ... \ ... \ , \ e^*_n \} $$
The relevant part of Winitzki's text reads as follows:http://mathhelpboards.com/attachments/linear-abstract-algebra-14/5348-dual-vector-space-dual-basis-another-question-winitzki-section-1-6-a-winitzki-dual-basis-pngIn the above quoted text from Winitzki, we read:
" ... ... Please note that $$e^*_1$$ depends on the entire basis $$\{ e^*_j \}$$ and not only on $$e^*_1$$, as might appear from the notation $$e^*_1$$. ... ... "I am puzzled by this statement ... can someone explain how, and indeed why, $$e^*_1$$ depends on the entire basis $$\{ e^*_j \}$$ and not only on $$e^*_1$$ ... a clarification of the nature of the dual basis would be most helpful ...
Hope someone can help ...
Peter===========================================================*** NOTE ***To indicate Winitzki's approach to the dual space (and its basis) and his notation I am providing the text of his introduction to Section 1.6 on the dual or conjugate vector space ... ... as follows ... ...
http://mathhelpboards.com/attachments/linear-abstract-algebra-14/5349-dual-vector-space-dual-basis-another-question-winitzki-section-1-6-a-winitzki-1-section-1-6-part-1-png
http://mathhelpboards.com/attachments/linear-abstract-algebra-14/5350-dual-vector-space-dual-basis-another-question-winitzki-section-1-6-a-winitzki-2-section-1-6-part-2-png