Determine vector spaces under the given operations

Question

Problems:

Given the set of objects and the operations of addition and scalar multiplication defined in each example below do the following:

Determine which sets are vector spaces under the given operations

For those sets that fail give the axiom(s) that fail to hold

1. The set of all triples of real numbers(x, y, z) with the operations defined

(x, y, z) + (x', y', z') = (x+x', y+y',z+z')

k(x, y, z) = (0, 0, 0)

2. The set of all pairs of real numbers(x, y) with the operations

(x, y) + (x', y') = (x+x', y+y')

k(x, y) = (2kx, 2ky)

3. The set of all pairs of real numbers(x, y) with the operations

(x, y) + (x', y') = (x+x'+1, y+y'+1)

k(x, y) = (kx, ky)

4. The set of all 2X2 matrices of the form

[^a ₁ ¹ _b]

With the standard matrix addition and scalar multiplication.

5. The set of all 2X2 matrices of the form

[^a _o ^o _b]

With the standard matrix addition and scalar multiplication.