Question

Let ?V be the set of vectors in ?2R2 with the following definition of addition and scalar multiplication:
Addition: [?1?2]⊕[?1?2]=[0?2+?2][x1x2]⊕[y1y2]=[0x2+y2]
Scalar Multiplication: ?⊙[?1?2]=[0??2]α⊙[x1x2]=[0αx2]
Determine which of the Vector Space Axioms are satisfied.

A1. ?⊕?=?⊕?x⊕y=y⊕x for any ?x and ?y in ?V
? YES NO

A2. (?⊕?)⊕?=?⊕(?⊕?)(x⊕y)⊕z=x⊕(y⊕z) for any ?,?x,y and ?z in ?V
? YES NO

A3. There exists an element 00 in ?V such that ?⊕0=?x⊕0=x for each ?∈?x∈V
? YES NO

A4. For each ?∈?x∈V, there exists an element  −?−x in ?V such that ?⊕(−?)=0x⊕(−x)=0
? YES NO

A5.  ?⊙(?⊕?)=(?⊙?)⊕(?⊙?)α⊙(x⊕y)=(α⊙x)⊕(α⊙y) for each scalar ?α and any ?x and ?y ?V
? YES NO

A6. (?+?)⊙?=(?⊙?)⊕(?⊙?)(α+β)⊙x=(α⊙x)⊕(β⊙x) for any scalars ?α and  ?β and any ?∈?x∈V
? YES NO

A7. (??)⊙?=?⊙(?⊙?)(αβ)⊙x=α⊙(β⊙x) for any scalars ?α and  ?β and any ?∈?x∈V
? YES NO

A8. 1⊙?=?1⊙x=x for all ?∈?x∈V
? YES NO

Answer 1