Linear space (geometry)
A linear space in geometry is like a playground where you can play with different toys that have numbers on them. Imagine you have a bunch of toys like balls, blocks, and cars with numbers on them. These numbers tell you how much you can move the toy in a certain direction. For example, if you have a ball with number '2' on it, it means you can move it two steps in a certain direction.
Now, in this playground, you can combine these toys to create new ones. For instance, you can combine two balls with numbers '2' and '3' on them to create a new ball with number '5' on it. This is called adding toys together.
You can also stretch or shrink these toys by multiplying them with a number. For example, if you have a car with number '4' on it and you double it, it becomes a car with number '8' on it. This is called scalar multiplication.
This playground has a set of rules that you need to follow when playing with these toys. These rules are called the axioms of linear space. Some of these rules are:
- Every toy has an opposite toy. For example, if you have a ball with number '2' on it, there is another ball with number '-2' on it that is its opposite.
- Adding two toys is commutative. This means that if you add a ball with number '2' on it to another ball with number '3' on it, it is the same as adding a ball with number '3' on it to a ball with number '2' on it.
- The playground has a special toy called the zero toy. This toy has number '0' on it and adding it to any other toy will give you the same toy back.
- Scalar multiplication is distributive over addition. This means that if you double a toy first and then add it to another toy, it is the same as adding the two toys first and then doubling the result.
Overall, a linear space is a place where you can play with toys that have numbers on them and follow certain rules to create new toys or stretch and shrink existing ones.
Now, in this playground, you can combine these toys to create new ones. For instance, you can combine two balls with numbers '2' and '3' on them to create a new ball with number '5' on it. This is called adding toys together.
You can also stretch or shrink these toys by multiplying them with a number. For example, if you have a car with number '4' on it and you double it, it becomes a car with number '8' on it. This is called scalar multiplication.
This playground has a set of rules that you need to follow when playing with these toys. These rules are called the axioms of linear space. Some of these rules are:
- Every toy has an opposite toy. For example, if you have a ball with number '2' on it, there is another ball with number '-2' on it that is its opposite.
- Adding two toys is commutative. This means that if you add a ball with number '2' on it to another ball with number '3' on it, it is the same as adding a ball with number '3' on it to a ball with number '2' on it.
- The playground has a special toy called the zero toy. This toy has number '0' on it and adding it to any other toy will give you the same toy back.
- Scalar multiplication is distributive over addition. This means that if you double a toy first and then add it to another toy, it is the same as adding the two toys first and then doubling the result.
Overall, a linear space is a place where you can play with toys that have numbers on them and follow certain rules to create new toys or stretch and shrink existing ones.
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