linear algebra help Let V = R^2 and define addition and scalar multiplication as

Question

linear algebra help

Let V = R^2 and define addition and scalar multiplication as follows u + v = (u_1, u_2) + (v_1, v_2) = (u_1 + v_1, 0) ku = k(u_1, u_2) = (ku_1, 0) Which of the following vector space axioms does not hold? (A) k(u + v) = ku + kv (B) 1u = u (C) Closure under scalar multiplication. (D) None of the above. Let V be a vector space, let k be a scalar, and let u, v, w be vectors in V. Which of the following statements does not hold? (A) (u + v) - w = u - (w - v) (B) If ku = 0, then k = 0 or u = 0. (C) -k(u + v - w) = -(ku - k(v - w)) (D) 0 - k(1u - v) + 0w = k(1v - u) + 0

Explanation / Answer

1)

B) does not hold as 1.u = (u1,0) which is not equal to u which is (u1,u2)

2)

C) does not hold as the property of scalar multiplication is distributive over addition but not subtraction

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