Non-Standard Addition & Multiplication in R^2: Zero Vector

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SUMMARY

The discussion centers on the non-standard definitions of addition and scalar multiplication in R², specifically defined as (x1,y1)+(x2,y2)=(x1,0) and c(x,y)=(cx,y). It concludes that these operations do not satisfy the axioms of a vector space, particularly highlighting the lack of commutativity and the absence of an additive identity. The proposed zero vector (0,-y1) fails to satisfy the equation u+0=u, confirming that the structure does not form a valid vector space.

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  • Understanding of vector spaces and their axioms
  • Familiarity with operations in R²
  • Knowledge of commutative properties in mathematics
  • Basic concepts of additive identities
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  • Study vector space axioms in detail
  • Explore examples of non-standard algebraic structures
  • Learn about commutative and associative properties in vector spaces
  • Investigate the implications of additive identities in mathematical structures
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Mathematicians, students of linear algebra, and anyone interested in the properties of vector spaces and non-standard mathematical operations.

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rather than use the standard definitons of addition and scalar multiplication in R^2, suppose these two operations are defined as follows.
(x1,y1)+(x2,y2)=(x1,0)
c(x,y)=(cx,y)

what would the zero vector be? can it be (0,-y1)? why or why not? I think it should fail this axiom u+0=u.
 
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skydiver_spike said:
rather than use the standard definitons of addition and scalar multiplication in R^2, suppose these two operations are defined as follows.
(x1,y1)+(x2,y2)=(x1,0)
c(x,y)=(cx,y)

what would the zero vector be? can it be (0,-y1)? why or why not? I think it should fail this axiom u+0=u.

First of all, your addition, as defined, is not commutative, i.e. (x1, y1) + (x2, y2) = (x1, 0) [itex]\neq[/itex] (x2, y2) + (x1, y1) = (x2, 0).
 
skydiver_spike said:
rather than use the standard definitons of addition and scalar multiplication in R^2, suppose these two operations are defined as follows.
(x1,y1)+(x2,y2)=(x1,0)
c(x,y)=(cx,y)

what would the zero vector be? can it be (0,-y1)? why or why not? I think it should fail this axiom u+0=u.

That is simply not a vector space. Not only is addition not commutative, as radou said, worse, there simply is NO additive identity. The equation (x1,y1)+ (x,y)= (x1,0)= (0,0) simply has no solution (x,y). This set is not a group with that addtion operation.
 

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