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Consider ordered pairs from R2 and define addition as: (x1,x2)+(y1, y2)=(x1+2y1, x2+2y2) and scalar multiplication as:
a(x1,x2)=(2ax1,ax2)
Show your work for each of the following.
(a) Does the above satisfy commutativity of addition:x+y=y+x?
(b) Does the above satisfy associativity of addition:(x+y)+z=x+(y+x)?
(c) Identity element of addition: ∃β=R^2 so that ∀x, x+β=x
(d) Does it satisfy the distributive law for scalar multiplication:∀a ∈R,a(x+y)=ax+ay?
(e) Identity element of multiplication: ∃a∈R so that ∀x,ax=x?
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