What is the significance of the zero vector in a vector space?

In summary: And if you think about it ... what makes the usual zero, "zero?" It's just that zero satisfies the properties of the zero element in the field of the real numbers. Other than that, it's just a point on the real line exactly like any other point. In summary, the zero vector in a vector space is the vector, a, such that, for any v in V, a\oplus v= v.
  • #1
74197
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Ok this is from a tutorial I am redoing again.

V5 = {(x, 1) | x ∈ R}, (x1, 1) ⊕ (x2, 1) := (x1 + x2, 1), c.(x, 1) := (cx, 1).

I understand that there exists a zero vector in this vector space, that comes in the form of (0,1). What I do not understand is why that is considered a zero vector for that vector space?

It is hard for me to 'see past' that 1 in the y-coordinate. Please ease my irritations, thank you :)
 
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  • #2
74197 said:
Ok this is from a tutorial I am redoing again.

V5 = {(x, 1) | x ∈ R}, (x1, 1) ⊕ (x2, 1) := (x1 + x2, 1), c.(x, 1) := (cx, 1).

I understand that there exists a zero vector in this vector space, that comes in the form of (0,1). What I do not understand is why that is considered a zero vector for that vector space?

It is hard for me to 'see past' that 1 in the y-coordinate. Please ease my irritations, thank you :)



According to what you defined [tex]\forall (x,1)\in V5\,\,,\,\,(x,1)\oplus (0,1)=(0,1)\oplus (x,1)=(x,1)[/tex] and thus that is the zero vector in that set (supposedly, a vector space).

DonAntonio
 
  • #3
The "zero vector" in any vector space, V, with vector addition [itex]\oplus[/itex] is the vector, a, such that, for any v in V, [itex]a\oplus v= v[/itex]. In other words, it is the "additive identity".

In this particular example, the "zero vector" is (0, 1): if v is any vector in this space, of the form (a, 1), then [itex](a, 1)\oplus (0, 1)= (a+0, 1)= (a, 1)=v[/itex].
 
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  • #4
Ah...I see now. It's not about the 'zero', it's more about the fact that the zero vector makes for an additive identity. Thank you Don and HoI :)
 
  • #5
74197 said:
Ah...I see now. It's not about the 'zero', it's more about the fact that the zero vector makes for an additive identity. Thank you Don and HoI :)

And if you think about it ... what makes the usual zero, "zero?" It's just that zero satisfies the properties of the zero element in the field of the real numbers. Other than that, it's just a point on the real line exactly like any other point.

That's what abstraction does ... it makes you see familiar things in a new way.
 

FAQ: What is the significance of the zero vector in a vector space?

What is a zero vector?

A zero vector is a vector with a magnitude of zero and no direction. It is represented by a vector with all components equal to zero, such as 0i + 0j + 0k in three-dimensional space.

What are the properties of a zero vector?

A zero vector has a magnitude of zero and no direction. It is also known as a null vector, and it is the only vector that is orthogonal to all other vectors.

Is the zero vector unique?

Yes, the zero vector is unique. It is the only vector with a magnitude of zero and no direction. Any other vector with these properties would be considered a zero vector.

What operations can be performed on a zero vector?

A zero vector can be added to other vectors, resulting in the original vector. It can also be multiplied by a scalar, resulting in a zero vector of the same dimension. However, it cannot be subtracted from other vectors or used as a divisor in vector division.

Can a zero vector have a direction?

No, a zero vector cannot have a direction. By definition, it has a magnitude of zero and no direction. Any attempt to assign a direction to a zero vector would result in a non-zero vector.

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