let e_1 and e_2 be the 2 vectors you gave. obviously e_1 and e_2
are lin. indep, so it's possible...just take orthogonal vectors
to the previous ones. But you could just use : e_3=[1 0 0 0] and
e_4=[0 0 1 0]...it's obvious that e_1,e_2,e_3 are pairwise
orthogonal, hence lin. indep.....combining e_2,e_3,e_4 as you
want will give you always a vector of the type [c b d b] hence
you cannot express e_1 as a combintation of the others, so the
e_i are lin. indep....clearly it's a generating subset of
R^4...so it's a basis. if f is a function on R^2, then f is a
function of 2 variables : f=f(x,y)...you gave only 1 variable
for the condition on f. 3) I think it's clear that applying a
permutation matrix on an n vector a=[a1...an] corresponds to a
permutation of the elements of the vector ? Apply a second
permutation matrice on b : c=M2*b since c is permutation of b
and b a permutation of a, then since the permuation are building
a group, c is a permutation of a..hence there exists a
permutation matrix M3 such that c=M