## help with vector spaces

1. The problem statement, all variables and given/known data
let $$S={(a_1,a_2):a_1,a_2 \in \mathbb{R}}$$ For $$(a_1,a_2),(b_1,b_2)\in{S}$$ and $$c\in\mathbb{R}$$ define $$(a_1,a_2)+(b_1,b_2)=(a_1+b_1,a_2-b_2)$$ and $$c(a_1,a_2)=(ca_1,ca_2)$$.
show that this is not a vector space

2. Relevant equations
vector space axioms

3. The attempt at a solution
this isn't an exercise in the book, but an example from the book that states that commutativity and associativity of addition and the distributive law all fail, so this in fact is not a vector space
i tried working these out and i think i got commutativity one right
because then you have $$(a_1+b_1,a_2-b_2)$$ does not equal $$(b_1+a_1,b_2-a_2)$$ is this correct?
i got stuck on associativity, i worked it out but to me it seems that it does in fact hold true
haven't check the distributive law though
the book im using is linear algebra by friedberg, insel and spence, second edition
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 Quote by miglo 1. The problem statement, all variables and given/known data let $$S={(a_1,a_2):a_1,a_2 \in \mathbb{R}}$$ For $$(a_1,a_2),(b_1,b_2)\in{S}$$ and $$c\in\mathbb{R}$$ define $$(a_1,a_2)+(b_1,b_2)=(a_1+b_1,a_2-b_2)$$ and $$c(a_1,a_2)=(ca_1,ca_2)$$. show that this is not a vector space 2. Relevant equations vector space axioms 3. The attempt at a solution this isn't an exercise in the book, but an example from the book that states that commutativity and associativity of addition and the distributive law all fail, so this in fact is not a vector space i tried working these out and i think i got commutativity one right because then you have $$(a_1+b_1,a_2-b_2)$$ does not equal $$(b_1+a_1,b_2-a_2)$$ is this correct?
Yes, that is correct.

 i got stuck on associativity, i worked it out but to me it seems that it does in fact hold true haven't check the distributive law though the book im using is linear algebra by friedberg, insel and spence, second edition
"Associativity of addition" would require that ((a1, b1)+ (a2, b2))+ (a3, b3)= (a1+ b1)+ ((a2, b2)+ (a3, b3)). What do you get for each of those?

Distributivity requires that (a1, b1)((a2, b2)+ (a3, b3))= (a1,b1)(a2,b2)+ (a1,b1)(a3,b3). What do you get for each of those?
 nevermind i just checked my work again and i think i figured it out for associativity we end up with $$(a_1+b_1+c_1,a_2-b_2-c_2)$$ on the left side while the right side gives us $$(a_1+b_1+c_1,a_2-b_2+c_2)$$ which aren't equal, so this shows associativity of addition fails right? im going to work on the distributive law, if i get stuck on that ill post back on this thread for help first time working with vector spaces, so im just trying to make sure i get this right before i actually start my linear algebra class in the fall haha

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## help with vector spaces

 Quote by miglo nevermind i just checked my work again and i think i figured it out for associativity we end up with $$(a_1+b_1+c_1,a_2-b_2-c_2)$$ on the left side while the right side gives us $$(a_1+b_1+c_1,a_2-b_2+c_2)$$ which aren't equal, so this shows associativity of addition fails right?
Yes, that is correct.

 im going to work on the distributive law, if i get stuck on that ill post back on this thread for help first time working with vector spaces, so im just trying to make sure i get this right before i actually start my linear algebra class in the fall haha
 thanks hallsofivy looking back into my book im not even sure if its the distributive law that fails my book says VS8 fails which is for a,b of elements in F(field) and each element x in V (a+b)x=ax+bx, is this the distributive law? it doesnt look like the one you posted hallsofivy anyways i checked it using x,y as the elements of the field $$(x+y)(a_1,a_2)=(xa_1+ya_1,xa_2+ya_2)$$ and $$x(a_1,a_2)+y(a_1,a_2)=(xa_1,xa_2)+(ya_1,ya_2)$$ both sides are not equal so it fails that axiom

 Tags axioms, laws, vector space

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