Question

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

Answer 1

Answer:

• The set $R^2$ with x ≤ 0 and y ≤ 0 is not a vector space.
• The 10th axiom  $cv \in V$ does not hold

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:

• 1. $u+v \in V$
• 2. v + w = w + v
• 3. (u + v) + w = u + (v + w)
• 4. v + 0 = v = 0 + v
• 5. v + (−v) = 0
• 6. 1v = v

• 7. c(dv) = (cd)v

• 8. c(v + w) = cv + cw
• 9. (c + d)v = cv + dv

• 10. $cv \in V$

Given the set of all vectors x, y in $R^2$ with x ≤ 0 and y ≤ 0

If the scalar c is such that $c<0, then cv\notin R^n as cv>0$.

Therefore, the 10th axiom is not satisfied and thus the set $R^2$ with x ≤ 0 and y ≤ 0 is not a vector space.