# Clarifying definitions of Euclidean and vector space.

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Starting my first few undergrad lectures this week and my geometry lecturer introduced vector space and euclidean space side-by-side in a very confusing (to me, at least) manner. After some thought, I have arrived at the following formal(-ish) definitions, and an intuitive link between the two. Can someone clear up any errors/misinterpretations please? I get the feeling these will be core to my course so I want to fully understand them from the start line.

n-dimensional vector space is the set of all n-tuple vectors together with the usual operations on vectors like scalar multiplication, vector addition, scalar/vector product.

n-dimensional euclidean space is the set of all points such that the distance is defined and a bijection exists from to such that each point is represented by a position vector.

intuitively, after some thought, it seems like euclidean space is just the collection of points onto which we project our vectors from vector space; the 'framework' in which we ground our notion of vector space.

I would be hugely grateful for any input you could give

n-dimensional vector space is the set of all n-tuple vectors together with the usual operations on vectors like scalar multiplication, vector addition, scalar/vector product.

n-dimensional euclidean space is the set of all points such that the distance is defined and a bijection exists from to such that each point is represented by a position vector.

intuitively, after some thought, it seems like euclidean space is just the collection of points onto which we project our vectors from vector space; the 'framework' in which we ground our notion of vector space.

I would be hugely grateful for any input you could give

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#2

Kind of.

Euclidean space is a specific example of a vector space. A vector space is a very general idea: any field alongside an appropriate set, with two operations "+" and "scalar multiplication" which in a certain sense behave nicely, is a vector space. For instance, is a vector space over , using as our addition the standard real-number + and as our scalar multiplication the standard real-number "times". The scalars are the rational numbers, and the vectors are single real numbers.

Euclidean space is a particular vector space ( over ), where the field is , the set is , and "+" and "scalar multiplication" are the usual. Notice that we also have a norm on this vector space (that is, your notion of distance); this is not a general feature of vector spaces, and that's what we mean by "the distance is defined". The dot product is also defined on this space, but again that's not a standard feature (in fact, having a dot product automatically gives you a norm in a particular way, and hence a distance).

Euclidean space is a specific example of a vector space. A vector space is a very general idea: any field alongside an appropriate set, with two operations "+" and "scalar multiplication" which in a certain sense behave nicely, is a vector space. For instance, is a vector space over , using as our addition the standard real-number + and as our scalar multiplication the standard real-number "times". The scalars are the rational numbers, and the vectors are single real numbers.

Euclidean space is a particular vector space ( over ), where the field is , the set is , and "+" and "scalar multiplication" are the usual. Notice that we also have a norm on this vector space (that is, your notion of distance); this is not a general feature of vector spaces, and that's what we mean by "the distance is defined". The dot product is also defined on this space, but again that's not a standard feature (in fact, having a dot product automatically gives you a norm in a particular way, and hence a distance).

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#3

(Original post by

n-dimensional vector space is the set of all n-tuple vectors together with the usual operations on vectors like scalar multiplication, vector addition,

**Ktulu666**)n-dimensional vector space is the set of all n-tuple vectors together with the usual operations on vectors like scalar multiplication, vector addition,

scalar/vector product.

This set of vectors, together with a rule for making a scalar/inner product, is called an inner product space. Once you have a scalar/inner product, then you can define the length (also called norm) of a vector in a standard way.

n-dimensional euclidean space is an inner product space built from the n-dimensional vector space . We take and add the usual "dot product" rule to create an inner product space. Since it's now an inner product space with this dot product, we can also define the length of vectors in it.

and a bijection exists from to such that each point is represented by a position vector.

intuitively, after some thought, it seems like euclidean space is just the collection of points onto which we project our vectors from vector space; the 'framework' in which we ground our notion of vector space.

intuitively, after some thought, it seems like euclidean space is just the collection of points onto which we project our vectors from vector space; the 'framework' in which we ground our notion of vector space.

There is a problem with the way vectors are taught at A level; it's not made clear that you can't automatically find the length of a vector without having defined an inner product. That confuses people when they come across the formal definition of a vector space for the first time. Everyone feels that there's something missing from the definition of a vector space, since there's nothing that allows you to measure the length of a vector without adding further rules.

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Thank you both for your responses. Right.. a general vector space is without the dot product / cross product, those are added later to create euclidean space (so in essence, euclidean space is a particular example of the more general vector space)? I was shown in the lecture how distance comes not from an inherent property of the vector space but from the dot product. He really didn't explain the difference between vector space and euclidean space well

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(Original post by

Thank you both for your responses. Right.. a general vector space is without the dot product / cross product, those are added later to create euclidean space (so in essence, euclidean space is a particular example of the more general vector space)? I was shown in the lecture how distance comes not from an inherent property of the vector space but from the dot product. He really didn't explain the difference between vector space and euclidean space well

**Ktulu666**)Thank you both for your responses. Right.. a general vector space is without the dot product / cross product, those are added later to create euclidean space (so in essence, euclidean space is a particular example of the more general vector space)? I was shown in the lecture how distance comes not from an inherent property of the vector space but from the dot product. He really didn't explain the difference between vector space and euclidean space well

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(Original post by

Yep, that's pretty much it. To get technical and boring, a vector space is nothing more nor less than a set X and a field F with operations "addition": X -> X, and "scalar multiplication": FxX -> X, such that under addition X is an abelian group, and such that , , , and . No mention of distance or dots at all.

**Smaug123**)Yep, that's pretty much it. To get technical and boring, a vector space is nothing more nor less than a set X and a field F with operations "addition": X -> X, and "scalar multiplication": FxX -> X, such that under addition X is an abelian group, and such that , , , and . No mention of distance or dots at all.

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