Question
Now we are ready to give the formal deﬁnition of a vector space. A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold:

[...]
commutativity
u + v = v + u for all u, v ∈ V;

associativity
(u+v)+w = u+(v +w) and (ab)v = a(bv) for all u,v,w ∈ V and all a,b ∈ F;

there exists an element 0 ∈ V such that v +0 = v for all v ∈ V;

for every v ∈ V, there exists w ∈ V such that v + w = 0;

multiplicative identity
1v = v for all v ∈ V;

distributive properties
a(u +v) = au +av and (a +b)u = au +bu for all a,b ∈ F and all u,v ∈ V.

Question
Now we are ready to give the formal deﬁnition of a vector space. A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold:

[...]
?

Question
Now we are ready to give the formal deﬁnition of a vector space. A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold:

[...]
commutativity
u + v = v + u for all u, v ∈ V;

associativity
(u+v)+w = u+(v +w) and (ab)v = a(bv) for all u,v,w ∈ V and all a,b ∈ F;

there exists an element 0 ∈ V such that v +0 = v for all v ∈ V;

for every v ∈ V, there exists w ∈ V such that v + w = 0;

multiplicative identity
1v = v for all v ∈ V;

distributive properties
a(u +v) = au +av and (a +b)u = au +bu for all a,b ∈ F and all u,v ∈ V.
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Now we are ready to give the formal deﬁnition of a vector space. A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold: commutativity u + v = v + u for all u, v ∈ V; associativity (u+v)+w = u+(v +w) and (ab)v = a(bv) for all u,v,w ∈ V and all a,b ∈ F; additive identity there exists an element 0 ∈ V such that v +0 = v for all v ∈ V; additive inverse for every v ∈ V, there exists w ∈ V such that v + w = 0; multiplicative identity 1v = v for all v ∈ V; distributive properties a(u +v) = au +av and (a +b)u = au +bu for all a,b ∈ F and all u,v ∈ V.

#### Original toplevel document (pdf)

owner: eshi - (no access) - Sheldon_Axler_Linear_Algebra_Done_Right.pdf, p23

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