# Proving If a set is a Vector Free Space

• maiad
In summary, the set of vectors defined by <a1,a2>+<b1,b2>=<a1+b1+1,a2+b2+1> and k<a1,a2>=<ka1+k-1,ka2+k-1> is not a vector space. This can be shown by the failure to satisfy the following axioms: commutativity of addition, associativity of addition, existence of a zero vector, existence of additive inverses, and distributivity of scalar multiplication. Further testing with concrete numbers may help in understanding why these axioms are not satisfied.
maiad

## Homework Statement

Determine whether set is a vector space. If not, give at least one axiom that is not satisfied.
the set of vectors <a1,a2>, addition and scalar multiplication defined by:
<a1,a2>+<b1,b2>=<a1+b1+1,a2+b2+1>
k<a1,a2>=<ka1+k-1,ka2+k-1>

## The Attempt at a Solution

For Vector Addition:
1) well since a1 and a2 is not restricted, the vector spaces are all real entities in V

2) Rule:x+y=y+x;
<a1,a2>+<b1,b2>=<a1+b1+1,a2+b2+1>=<b1,b2>

3)Rule:(x+y)+z=(x+(y+z);
(<a1,a2>+<b1,b2>)+<0,0>= <a1,a2>+(<b1,b2>+<0,0>)

4)Not sure how to prove is this set has a unique vector O in V such that O+x=x+O

5)Rule: There exist a vector where x+(-x)=(-x)+x=O;
Since the set is not restricted, there exist a negative vector where a1+(-a1)=(-a1)+a1

For Scalar Multiplication:
6)Since the set is no restricted, any scalar would be in the space

7)Rule:k(x+y)=kx+ky;
Not sure how to prove this one...

8)Rule: (k1+k2)x=k1x+k2x
nor this one

9)Rule:k1(k2x)=(k1k2)x
or this one

10)Rule: 1x=x
This one obviously satisfies

Can someone give me hints on the ones i didn't get and also see if the ones i did is correct?

Do not assume the space satisfies all your axioms. Try the ones you are not sure about with some concrete numbers. 2<3,5> and such.

You left of an important part of 4). 0+x = x+0 = x

BTW, When you say something is "obvious" you should double check yourself. Most errors occur in the "obvious" cases because we stop considering them too soon. (I'm not saying you made an error. This is just good general advice.)

## What is a vector free space?

A vector free space, also known as a vector space, is a set of mathematical objects called vectors that can be added and multiplied by a scalar, such as a number, to create new vectors. This set must also follow specific rules and properties to be considered a vector free space.

## What are the rules for a set to be considered a vector free space?

To be considered a vector free space, a set must meet the following conditions:

• The set must contain at least one vector.
• The set must be closed under vector addition and scalar multiplication.
• The set must have an additive identity (a vector that when added to any other vector, results in that vector).
• The set must have an additive inverse (a vector that when added to another vector, results in the additive identity).
• The set must have a multiplicative identity (a scalar that when multiplied by any vector, results in that vector).
• The set must follow the commutative, associative, and distributive properties.

## How can I determine if a set is a vector free space?

To determine if a set is a vector free space, you must check if it meets all of the above conditions. This can be done by performing various operations on the vectors in the set and checking if the results follow the rules and properties of a vector free space. If the set fails to meet even one of the conditions, it cannot be considered a vector free space.

## What is the importance of proving if a set is a vector free space?

Proving if a set is a vector free space is important because it allows us to understand the properties and behavior of the vectors in that set. This can help us solve complex mathematical problems and make predictions in various fields such as physics, engineering, and computer science. It also provides a foundation for further study and application of vector spaces.

## What are some real-world applications of vector free spaces?

Vector free spaces have numerous real-world applications, some of which include:

• In physics, vector free spaces are used to represent forces, velocities, and other physical quantities.
• In computer graphics, vector free spaces are used to represent 2D and 3D objects and manipulate them for animation and simulation.
• In finance, vector free spaces are used to model and predict stock prices and other economic variables.
• In machine learning, vector free spaces are used to represent and analyze data for tasks such as image and speech recognition.

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