Clarifying definitions of Euclidean and vector space.

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

\mathbb{R}^n

is the set of all n-tuple vectors

(v_1 , v_2 , \cdots , v_n)

together with the usual operations on vectors like scalar multiplication, vector addition, scalar/vector product.

n-dimensional euclidean space

\mathbb{E}^n

is the set of all points

P , Q , \cdots

such that the distance

|PQ|

is defined and a bijection exists from

\mathbb{E}^n

\mathbb{R}^n

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 :smile:

Reply 1

Smaug123

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,

\mathbb{R}

is a vector space over

\mathbb{Q}

, 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 (

\mathbb{R}^n

over

\mathbb{R}

), where the field is

\mathbb{R}

, the set is

\mathbb{R}^n

, 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).

Reply 2

atsruser

Original post by Ktulu666

n-dimensional vector space

\mathbb{R}^n

is the set of all n-tuple vectors

(v_1 , v_2 , \cdots , v_n)

together with the usual operations on vectors like scalar multiplication, vector addition,

Yes. A vector space is a set of objects which you can add up and scale like the vectors you know from mechanics and physics. The numbers that you scale by in

\mathbb{R}^n

are the usual real numbers. That's all that is defined in a vector space. In a vector space there is also defined a 0 vector and the negative of a vector, which work as you would expect when added to other vectors.

scalar/vector product.

No. These aren't defined in a vector space. If you want a scalar product, then you need to define a new operation on your vectors (called the scalar or inner product of two vectors), then add that as a new rule to the ones you already have for a vector space.

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

\mathbb{E}^n

is the set of all points

P , Q , \cdots

such that the distance

|PQ|

is defined

n-dimensional euclidean space is an inner product space built from the n-dimensional vector space

\mathbb{R}^n

. We take

\mathbb{R}^n

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

\mathbb{E}^n

\mathbb{R}^n

I don't follow much of this, I'm afraid.

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.

Reply 3

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 :s-smilie:

Reply 4

Smaug123

Original post by Ktulu666

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

\lambda (x + y) = \lambda x + \lambda y

(\lambda+\mu)x = \lambda x + \mu x

1 x = x

, and

a (b x) = (ab) x

. No mention of distance or dots at all.

Reply 5

Ktulu666

Original post by Smaug123

\lambda (x + y) = \lambda x + \lambda y

(\lambda+\mu)x = \lambda x + \mu x

1 x = x

, and

a (b x) = (ab) x

. No mention of distance or dots at all.

Thanks. I think I have a vague enough picture in my head to be able to work with now. I don't think I've ever met anything quite so abstract before... After speaking to my tutor during one of my tutorials he told me not to get too caught up on the distinction, so my lack of a crystal clear 'intuitive definition' doesn't phase me as much now. Thank you all again for your help :smile: