While self studying Linear Algebra from Hoffman and Kunze, I have following question.
I have a question in blue highlighted line ( Page 103 of book)
How does in the underlined line, n- tuple gives the co-ordinate of $f_{i} $ "relative to basis which is dual to standard basis for $F^{n} $ ? .
Clarly, by first equation on Pg 103 $f_{i} $ can be written as in terms of $A_{ij}$ , j belonging from 1 to n. But how the co-ordinnates are " relative to basis which is dual to standard basis of $F^{n} $ ?
Kindly explain this line?
Write $e_1^*,\ldots,e_n^*$ for the basis dual to the standard basis.
If we write $x=(x_1,\ldots,x_n)$, then $x_j=e^*_j(x)$ for all $1\leq j\leq n$. So $$ f_i(x)=\sum_{j=1}^nA_{ij}x_j=\sum_{j=1}^nA_{ij}e^*_j(x) $$ for all $x$, whence $$ f_i=\sum_{j=1}^nA_{ij}e_j^*. $$