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

However, the set {(x,0)} IS a vector space with that addition and the given scalar multiplcation. To see that that is true, let (x,y) be any vector in R^2 and let (x1,0) be any vector in R^2. In summary, the given operations of addition and scalar multiplication for R^2 do not form a vector space as the addition is not commutative and there is no additive identity. However, the set {(x,0)} is a vector space with these operations.
  • #1
skydiver_spike
1
0
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.
 
Physics news on Phys.org
  • #2
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).
 
  • #3
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.
 

1. What is a zero vector in R^2?

A zero vector in R^2 is a vector with both of its components equal to zero. In other words, it has no magnitude or direction and is represented by the point (0,0) in the coordinate plane.

2. Why is the zero vector important in non-standard addition and multiplication in R^2?

The zero vector serves as the identity element for addition and multiplication in R^2. This means that when the zero vector is added to or multiplied by any other vector, the result is that same vector. It also helps to maintain the closure property, as any vector added to or multiplied by the zero vector will still be a vector in R^2.

3. Can the zero vector be added to or multiplied by any vector in R^2?

Yes, the zero vector can be added to or multiplied by any vector in R^2. However, the resulting vector will be the same as the original vector and will not change in magnitude or direction.

4. Is the zero vector unique in R^2?

Yes, the zero vector is unique in R^2. This means that there is only one zero vector in R^2 and it cannot be changed or transformed into any other vector.

5. How is the zero vector represented in R^2?

The zero vector is represented by the point (0,0) in the coordinate plane. It can also be represented using vector notation as 0.

Similar threads

I Vector subspace and basis vectors in the context of data science
    • Linear and Abstract Algebra
Replies
6
Views
873
I Proving Convexity of the Set X = {(x, y) E R^2; ax + by <= c} in R^2
    • Linear and Abstract Algebra
Replies
5
Views
1K
I When is a subset a subspace in a vector space?
    • Linear and Abstract Algebra
Replies
7
Views
2K
MHB Proving Vector Subspaces of R^3
    • Linear and Abstract Algebra
Replies
5
Views
1K
I Showing that a set of differentiable functions is a subspace of R
    • Linear and Abstract Algebra
Replies
19
Views
4K
MHB Determine if vector space.
    • Linear and Abstract Algebra
Replies
5
Views
1K
I Prove $$T_{p}M$$ is a vector space with the axioms
    • Differential Geometry
Replies
3
Views
1K
MHB Proving Vector Subspace $U \notin R^3$
    • Linear and Abstract Algebra
Replies
2
Views
1K
MHB Determine vec {{x},{y},{3x+2y}} in R^3 form a vec space
    • Linear and Abstract Algebra
Replies
11
Views
2K
MHB Are $V_1$ and $V_2$ Vector Spaces According to Defined Properties?
    • Linear and Abstract Algebra
Replies
14
Views
3K

Hot Threads

Recent Insights

Back
Top