Complete the set S = {x³+x²+1,x²+x+1,x+1} to get a basis of P₃

Let V = R³ andW = {(x,y, z)|x+y+z = 0} be a subspace of V. Which of the
following pairs of vectors are in the same coset of W in V?
i) (1,3,2) and (2,2,2). ii) (1,1,1) and (3,3,3).

Reduce the conic x²+6xy+y2 - 8 = 0 to standard form. Hence identify the given conic.

Check whether each of the following subsets of R³ is linearly independent.
i) {(1,2,3), (-1,1,2), (2,1,1)}. ii) {(3,1,2); (-1,-1,-3), (-4,-3-0)}.

Find the orthogonal canonical reduction of the quadratic form
x²+y² + z² - 2xy - 2xz - 2yz. Also, find its principal axes.

Let V = R³, A = {(x;y; z)|y = 0} and B = {(x,y, z)|x = y = z}. Check whether R³ = A ⊕ B