Determining if certain sets are vector spaces

In summary, the homework statement states that the set of all pairs of real numbers (1,x) with the operations (+,-,*), and k(1,x)=(1,kx) is a vector space. The Attempt at a Solution states that the additive identity is (1,0), and if this is not correct, then the equation holds true. However, the equation is confusing because it is written in a way that might confuse the reader.
  • #1
csc2iffy
76
0

Homework Statement



The set of all pairs of real numbers of the form (1,x) with the operations:
(1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx) k being a scalar

Is this a vector space?

Homework Equations


(1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx)

The Attempt at a Solution


I verified most of the axioms hold, but I'm unsure about the additive identity.
Can it be something other than the zero vector?
My attempt, "O" being the additive identity
O=(1,0)
A+O=(1,x)+(1,0)=(1,x)
This means the additive inverse must equal (1,0)
A+(-A)=(1,x)+(-1,-x)=(1,x+(-x))=(1,0)
If this isn't right, then I know it doesn't hold. I'm just a little confused. Thanks for any help
 
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  • #2
That's all correct. You understand this very well. They defined the operations in a way that was maybe intended to confuse you. But you weren't confused. (1,0) is the 'zero' vector.
 
  • #3
If A=(1,x) then -A is (1,-x) not (-1,-x). Otherwise it looks like you've got it. You can imagine just chopping off the 1 here and describing the element (1,x) by just the number x. Then what you really have is just the standard real numbers
 
  • #4
Office_Shredder said:
If A=(1,x) then -A is (1,-x) not (-1,-x). Otherwise it looks like you've got it. You can imagine just chopping off the 1 here and describing the element (1,x) by just the number x. Then what you really have is just the standard real numbers

Ooops. Missed that. Thanks for being careful.
 
  • #5
csc2iffy said:

Homework Statement



The set of all pairs of real numbers of the form (1,x) with the operations:
(1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx) k being a scalar

Is this a vector space?


Homework Equations


(1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx)


The Attempt at a Solution


I verified most of the axioms hold, but I'm unsure about the additive identity.
Can it be something other than the zero vector?
My attempt, "O" being the additive identity
O=(1,0)
A+O=(1,x)+(1,0)=(1,x)
This means the additive inverse must equal (1,0)
A+(-A)=(1,x)+(-1,-x)=(1,x+(-x))=(1,0)
If this isn't right, then I know it doesn't hold. I'm just a little confused. Thanks for any help
Sort of OK. Besides the correction above, the wording needs a little work.

The sentence "This means the additive inverse must equal (1,0) ." should be changed to something like: "This means when the additive inverse of any number pair is added to that number pair, the result is (1,0)."
 
  • #6
Thank you all for the very quick responses! That makes me feel so much better... I have a final coming up on this :)
 

FAQ: Determining if certain sets are vector spaces

1. What are the criteria for a set to be considered a vector space?

To be considered a vector space, a set must satisfy the following criteria:

  • It must contain a zero vector.
  • It must be closed under vector addition.
  • It must be closed under scalar multiplication.
  • It must have an associative and commutative addition operation.
  • It must have a distributive property between scalar multiplication and vector addition.
  • It must have an identity element for scalar multiplication.

2. Why is it important to determine if a set is a vector space?

Determining if a set is a vector space is important because it allows us to apply properties and operations from vector spaces to the set. This can help with solving problems and making calculations easier.

3. Can a set be a vector space if it does not have a zero vector?

No, a set cannot be considered a vector space if it does not contain a zero vector. The zero vector is an essential component of a vector space and without it, the set would not satisfy the criteria for being a vector space.

4. How can we prove that a set is a vector space?

To prove that a set is a vector space, we must show that it satisfies all the criteria for being a vector space. This can be done by demonstrating that the set contains a zero vector, is closed under vector addition and scalar multiplication, and follows the necessary properties such as associativity, commutativity, and distributivity.

5. Is a vector space unique or can it have multiple forms?

A vector space can have multiple forms as long as it satisfies the criteria for being a vector space. For example, the set of all real numbers can be a vector space in its standard form, but it can also be a vector space in its polynomial form. Both forms satisfy the criteria for being a vector space.

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