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

• Pushoam
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.
Pushoam

## Homework Equations

definition of null vector,
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## 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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Last edited:
Yes, or even easier ##(0,0,0) \notin \{\,(a,b,1)\,\}##.

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

Thanks for it.

## 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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