Flawed Vector Space Example: Showing Failures of Commutativity and Associativity

[miglo]

Homework Statement

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

Homework Equations

vector space axioms

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 I am using is linear algebra by friedberg, insel and spence, second edition

 

[HallsofIvy]

Homework Statement

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

Homework Equations

vector space axioms

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 I am 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?

 

[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?

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 I am just trying to make sure i get this right before i actually start my linear algebra class in the fall haha

 

[HallsofIvy]

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 I am just trying to make sure i get this right before i actually start my linear algebra class in the fall haha

 

[miglo]

thanks hallsofivy

looking back into my book I am 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 doesn't 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