# simple explanation of a vector space

#### PG1995

Joined Apr 15, 2011
813
Hi, I have some basic knowledge of matrices and vectors. mI understand some of their practical applications in real life. This idea of vector is entirely new to me. What is a vector in simpl terms, or to begin with? Please try to explain with some simple practical explanation. I would really appreciate this vector space teaching of yours. Many thanks.

#### TBayBoy

Joined May 25, 2011
145
Here is a PowerPoint presentation that might help.

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#### Wendy

Joined Mar 24, 2008
21,848
Can't view it, my Office is too old. Every use compatibility modes? The ones without the x at the end (.pptx vs .ppt).

#### TBayBoy

Joined May 25, 2011
145
Unfortunately I'm not at school right now, so can't convert it (out for the summer )

#### steveb

Joined Jul 3, 2008
2,431
• TBayBoy

#### johnmtb

Joined Sep 26, 2009
1
All,

what am i doing wrong?

i wanted to download the attached files, but i am told that i am not logged in. if i wasn't logged in i wouldn't be able to post.

what do i do now?

regards,

john

Joined Jul 7, 2009
1,577
The concept of a vector space is one of the most useful things you'll study, as it has far-reaching ramifications and applications. A vector space is a collection of vectors. This collection has properties required by the axioms of a vector space. The axioms cover things like how two vectors of the space can be combined (in various ways) to yield a third vector, also in the space. There are also ways of combining vectors such that they make objects that are no longer vectors; these are studied in a variety of different "algebras" you may come across in your future studies.

The axioms of a vector space are pretty "mild", but they yield rich structures with lots of useful properties -- which is why they are worth studying. Since they are used for modeling a variety of things in technology, these models then have the useful properties of the vector space. And, of course, they're only really useful if these properties mirror the things we see in the real world. They do, which is why vector spaces have survived in a Darwinian sense in the school curricula and the various technical fields.

The use of the term "space" is a clue to the origin of vector spaces -- they were used to study the properties of physical space. In such cases, concepts like nearness and distance become useful and are well-defined.

Then comes the term "vector". So what is a vector? It's a collection of some things. Now, this is also the informal concept of a set. The vector is distinguished from the set in that the order of the elements given is important. Thus, if {a1, a3, a2} are a set, order of the elements is unimportant and you can equally well denote the set as {a2, a3, a1}. A vector is usually denoted with parentheses and (a1, a2, a3) is a different vector from (a2, a3, a1). The elements a1, a2, ... all share some characteristic(s) which make them a suitable for inclusion in a vector and vector space. Most often, the elements are numbers, usually real numbers or complex numbers.

In elementary technical studies, these numbers are often spatial coordinates -- like the position of a point mass in physics in three dimensional space. But it's important to realize that the machinery (lemmas and theorems) of vector spaces apply to any vector -- and in more exotic technical things, the components of the vectors might not be our familiar numbers anymore. But if these vectors satisfy the axioms of a vector space, you can still use the machinery on them. This is an example of the power of abstraction in mathematics. Then, they can be combined with other mathematical things to yield other useful things. If I was 20 years old again and had more than one functioning neuron, I'd love to study more about all these things... #### doinky

Joined Jul 19, 2011
12
Sounds like were headed into the twilight zone.

#### PG1995

Joined Apr 15, 2011
813
It seems like I made a serious mistake in my original post which was made more than a year ago. At that time probably I was too busy that I didn't pursue the topic any further. Nonetheless, I'm thankful to everyone who tried to help me. Now reading someonesdad's post again makes more sense.

In my original post I missed the word "space" at important places which probably let many readers to believe that I was mainly asking about vectors.

I have some basic knowledge of matrices and vectors. I understand some of their practical applications in real life. This idea of vector space is entirely new to me. What is a vector space in simple terms, or to begin with?
I'm of opinion that mathematics is well understood when its applications are seen in physical sciences such as physics. It actually shows how mathematics models real world phenomena. For example, in a dictionary you can devote quite a few pages to detail what a cat or dog looks like but all that detail wouldn't be much useful until one sees a real cat or dog. But all that detail would become very useful once there is a 'visual picture' of a concept.

The concept of a vector field is quite easy to grasp if one has an understanding of gravitational, electric or magnetic field (I hope I'm not wrong in saying so). Likewise, a scalar field is easy to deal with if one has an understanding of electric or gravitational potential.

Actually I'm still struggling with concepts of vector space and subspace. Is a vector space related to a vector field? Could you relate the mathematical concept of vector space to something 'concrete' the way, for example, I have related vector field concept with electric or magnetic field? Thank you.

Regards
PG

#### bretm

Joined Feb 6, 2012
152
They're different things. A vector field, like an electric field, defines a vector at each point in space.

A vector space is defined by a set of vectors. Any point that can be represented by the sum of those vectors, each scaled independently, belongs to that space.

For example, a single vector (1, 0, 0) by itself defines a vector space containing only the points on the x axis, because every point on the x axis is a scalar multiple of that vector.

Add a second vector (0, 1, 0), and now they form a vector space that contains all points on the xy plane. Points with z other than zero are not in the space, because they can't be represented as a sum of scaled versions of those two vectors.

Those vectors are called the basis of that vector space, but the basis is not unique for that space. The pair of vectors (1,2,0) and (2,-1,0) defines the same vector space (all points in xy plane).

• PG1995

#### PG1995

Joined Apr 15, 2011
813
A vector space is defined by a set of vectors. Any point that can be represented by the sum of those vectors, each scaled independently, belongs to that space.

For example, a single vector (1, 0, 0) by itself defines a vector space containing only the points on the x axis, because every point on the x axis is a scalar multiple of that vector.

Add a second vector (0, 1, 0), and now they form a vector space that contains all points on the xy plane. Points with z other than zero are not in the space, because they can't be represented as a sum of scaled versions of those two vectors.
Thank you.

From a layman's point of view, I would say if x-axis taken to be a 'line' vector space, then the vector (1, 0, 0) is an element of that space. The same goes for the other vector (0, 1, 0) which is an element of y-axis vector space.

The following videos could be of help to someone like me.

1: http://www.youtube.com/watch?v=f6KGArgXhzs (introduction of a vector space)
2: http://www.youtube.com/watch?v=LCmJ3TgCBSQ&feature=relmfu (properties of a vector space)