Show: Vectors e.g.(a,b,1) do not form vector space.

In summary, a vector space is a mathematical structure consisting of vectors and operations such as addition and scalar multiplication. When vectors do not form a vector space, it means they do not meet all necessary properties. It is important for vectors to form a vector space because they have real-world applications and provide a framework for understanding and manipulating them. Examples of vectors that do not form a vector space include those with different dimensions or without a zero vector or inverses. While these vectors can still be useful in certain situations, it is important to recognize when they do not form a vector space to avoid errors.
  • #1
Pushoam
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Homework Statement


upload_2018-8-12_19-29-5.png


Homework Equations


definition of null vector,
upload_2018-8-12_19-40-16.png
[/B]

The Attempt at a Solution


null vector : ## |0 \rangle = (0,0,0) ##
inverse of (a,b,c) = ( - a, -b, -c)
vector sum of the two vectors of the same form e.g. (c,d,1) + ( e,f,1) = ( c+e, d+f, 2) does not have the same form. So, the vectors of the form ( a,b,1) do not form a vector space.

Is this correct?
 

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  • #2
Yes, or even easier ##(0,0,0) \notin \{\,(a,b,1)\,\}##.
 
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Likes Pushoam
  • #3
fresh_42 said:
Yes, or even easier ##(0,0,0) \notin \{\,(a,b,1)\,\}##.

Thanks for it.
 

Related to Show: Vectors e.g.(a,b,1) do not form vector space.

What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and operations that can be performed on those vectors. These operations include addition and scalar multiplication, which allow for the creation of new vectors within the space.

What does it mean for vectors to not form a vector space?

When vectors do not form a vector space, it means that they do not satisfy all of the necessary properties of a vector space. These properties include closure under addition and scalar multiplication, existence of a zero vector, and existence of additive and multiplicative inverses.

Why is it important for vectors to form a vector space?

Vector spaces are important in mathematics and science because they provide a framework for understanding and manipulating vectors. They also have many real-world applications, such as in physics, engineering, and computer graphics.

What are some examples of vectors that do not form a vector space?

Examples of vectors that do not form a vector space include a set of vectors with different dimensions, a set of vectors where the operations do not result in a vector within the set, and a set of vectors that do not have a zero vector or additive and multiplicative inverses.

Can vectors that do not form a vector space still be useful?

Yes, vectors that do not form a vector space can still have practical applications. For example, in physics, vectors may not always form a vector space but can still be used to represent quantities such as force or velocity. However, it is important to recognize when vectors do not form a vector space in order to avoid mathematical errors.

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