Understanding Vector Spaces: Properties and Applications

[member 392791]

Hello,

I am wondering why is it that matrices and infinite sequences may be considered part of a vector space. I have read 3 different sources, and my interpretation of a vector space is something that belongs in a field and follows a list of properties that are standard to real numbers, i.e association, commutativity, zero property etc. It must have closure by addition and scalar multiplication, as well as being a nonempty set.

Is the reason that a matrix can be included in a vector space is that it can be multiplied to a vector to give a constant. I think this would make sense since matrices follow the properties listed above, but how linear equations exist in a real number space pervades me, perhaps it is similar to a straight line existing in an xyz-coordinate system.
Ax = b where x is a vector

How is a vector space different from a typical coordinate system, other than it can go into higher dimensions?

 

[chiro]

Hey Woopydalan.

There is infinite-dimensional vector spaces and infinite-dimensional operators and the first can be found under the study of Hilbert-Spaces and the second can be found under the study of operator algebra's:

http://en.wikipedia.org/wiki/Hilbert_space

http://en.wikipedia.org/wiki/Operator_algebra

A Hilbert-Space is a complete inner product space (which is also continuous) and a Banach Space is a complete normed space (which is also continuous in the norm).

 

[tiny-tim]

Hello Woopydalan!

… how linear equations exist in a real number space pervades me, perhaps it is similar to a straight line existing in an xyz-coordinate system.
Ax = b where x is a vector

Basically, if you can add things, and if you can multiply them by a scalar (and you can do both those for linear equations), then it's a vector space.

How is a vector space different from a typical coordinate system …

Not sure what you mean by a typical coordinate system, but if it's what i think, then every linear typical coordinate system is a vector space, and vice versa.

 

[Studiot]

A vector space is a set whose members satisfy the vectors space axioms and are called vectors.

A vector space is not a field. Scalars are members of a second set which is a field.

One of the important properties of the axioms is that the operations defined will always get you another member of the set and work for each and every member of the set. Further there is always a suitable member to perform these operations.

That is for every a + b there is always a c in the set.

Note that the basic axioms only include one operation between vectors, called addition of vectors. This one is mandatory.

Some vector spaces have other operations, such as multiplication of vectors etc.

A good list of the axioms for your purposes is at

http://www.math.ucla.edu/~tao/resource/general/121.1.00s/vector_axioms.html

 

[member 392791]

Ok a separate issue, I am trying to understand subspaces better. They are saying in the book if a vector v is a linear combination of the elements of the vector space V, then is it the case that v spans V?

If v is a linear combination of u, and u is a combination of the elements of V, then U is V. These terminologies are confusing me

 

[tiny-tim]

… u is a combination of the elements of V …

what does that mean?

anyway, a single vector can only span (or, more correctly, generate) a one-dimensional subspace

 

[Studiot]

First subspaces.

Subspaces are also vector spaces.

That is they are complete or obey the property I highlighted before, ie they contain all the vectors of a particular type and you can always find a c for any a+b.

As a for instance

Any plane is a subspace of the threeD vector space we use in geometry.

Take the X-Y plane ; all the vectors of the form αX+βY live in this plane. There are no vectors that have this form that do not live in this (sub)space.

Now we say the general vector αX+βY with α,β ≠ 0 spans the subspace because the X-Y plane is the smallest (sub)space that can contain such vectors. (A span is the smallest set that satisfies the given conditions. You may meet the idea in other contexts).
We have the non zero restriction because if say β = 0 then the vector is αX+0Y = αX.
This vector does not span the X-Y subspace since it contains no information about vectors with a Y value. You can take it that a spanning vector has a non zero value for every coordinate axis.

So yes if v is a linear combination of all the members of V (except itself) then v spans V .

Sorry if this is a bit rambling but you caught me on the hop.