Say whether the following sets are vector subspaces of the vector space V in each of the following cases: 1) S1 = {(x, y) ∈ R 2 : |x| = |y|} , V = R 2

xplain how one can check whether a subset of a space is a subspace. Consider two sets{(x, y) ∈ R2 | |x| = |y|} and {(x, y) ∈ R2 | |x| = −|y|}. One of them is a subspace of R2 and theother is not. Using any method you prefer, decide which set is a subspace of R2 and which is not.

Which of the following are subspaces of R3 ?(i) {(x,y,z)| z = 2x+3y+2}(ii) {(x,y,z)| x2+y2=z2}Select one:a. (ii) onlyb. (i) and (ii)c. None of the other choices is correctd. (i) only

Which of the following subsets of R3 are subspaces?(a) The plane of vectors (b1, b2, b3) with b1 = b2.(b) The plane of vectors with b1 = 1.(c) The vectors with b1b2b3 = 0.

Determine whether the set S spans R2. If the set does not span R2, then give a geometric description of the subspace that it does span.S = {(−1, 2), (2, −1), (1, 1)}S spans R2.S does not span R2. S spans a line in R2.    S does not span R2. S spans a point in R2.

Determine whether the given set, together with the specified operations of addition and scalar multiplication, is a vector space. If it is not, select all of the axioms that fail to hold. (Let u, v, and w be vectors in the vector space V, and let c and d be scalars.)The set of all vectors x y in ℝ2 with xy ≥ 0 (i.e., the union of the first and third quadrants), with the usual vector addition and scalar multiplication