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vampirchik [111]
3 years ago
8

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
Mathematics
1 answer:
oksano4ka [1.4K]3 years ago
6 0

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 c0.

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

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