a) since each is a real number, isn't it true that ? you also need to check linearity in x and y.
5)there is which is defined by where
A)check if this function is a simetric bilenear.
B)find the representative matrices by the standart basis of
C)find the jordan form of the matrices you found in part B
D)check if for every
??
regarding A:
f is simetric if f(x,y)=f(y,x) so i need to prove that
but when i put the given and i get two different expressions
?
i made a typing mistake in the questions
ok i understand part A
regarding B)
for standart basis (1,0..,0) .. (0,0..,1)
so for each vector in this basis
and
in transformations each column of the representation matrices would be T(e1)
what to do here
first of all, your sum isn't correct. if you want to calculate f(ei,ej) you should sum over some other index, like k (because k can be anything from 1 to n, but i and j are "constants").
for k ≠ i,j, we have the term 1/n^2, for k = i or j, we have the term (1 - 1/n)(-1/n) = (1-n)/n^2 (when i ≠ j), so that f(ei,ej) = n[(n-2)/n^2 + (2-2n)/n^2] = -n/n = -1.
when i = j, we have n[(n-1)/n^2 + (n-1)^2/n^2] = n-1. so for n = 2, for example, you should get the matrix:
[1 -1]
[-1 1].