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 x ≤ 0, y ≤ 0 (i. e., the third quadrant), with the usual vector addition and scalar multiplication

Step-by-step explanation:

Let u, v and w be vectors in the vector space V, and let c and d be scalars. A set S is defined as a vector space if it satisfies the following conditions:

Given the set of all vectors x, y in with x ≤ 0 and y ≤ 0

If the scalar c is such that .

Therefore, the 10th axiom is not satisfied and thus the set with x ≤ 0 and y ≤ 0 is not a vector space.

in geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. by extension, a line and a plane, or two planes, in three-dimensional euclidean space that do not share a point are said to be parallel.