6.1. Decide whether the given set of vectors is linearly independent in the indicated vector space.
a. { x^4 + x^3 + x^2 + x + 1, x^2 + x + 2, x } in R[x] over R,
b. { (1,0,1), (1,1,0), (0,1,1) } in Z_2^3 over Z_2,
c. { (1,0,1), (1,1,0), (0,1,1) } in R^3 over R,
d. { sin x, cos x, x } in R^R over R,
e. { sin^2 x, cos^2 x, sin 2x, cos 2x } in R^R over R,
f. { x_1, x_1 + x_2, x_1 + x_2 + x_3, ..., x_1 + ... + x_n } if { x_1, x_2, x_3, ..., x_n } is linearly independent, in any vector space V,
g. { x_1 + x_2, x_2 + x_3, ..., x_{n-1} + x_n, x_n + x_1 } if { x_1, x_2, x_3, ..., x_n } is linearly independent, in any vector space V,
h. { x_1 + v, x_2 + v, ..., x_n + v } if { x_1, x_2, x_3, ..., x_n } is linearly independent and v is any vector,
i. S, where S ⊆ T and T is linearly independent,
j. S, where T ⊆ S and T is linearly independent,
k. { x_1, x_2, x_3, ..., x_n }, where x_i ∈ span(S_i), x_i ≠ Θ for i=1,2, ..., n, sets S_i are finite and pairwise disjoint and the set S_1 ∪ S_2 ∪ ... ∪ S_n is linearly independent.