Continue to Site

Welcome to our site!

Electro Tech is an online community (with over 170,000 members) who enjoy talking about and building electronic circuits, projects and gadgets. To participate you need to register. Registration is free. Click here to register now.

  • Welcome to our site! Electro Tech is an online community (with over 170,000 members) who enjoy talking about and building electronic circuits, projects and gadgets. To participate you need to register. Registration is free. Click here to register now.

simple explanation of a vector space

Status
Not open for further replies.

PG1995

Active Member
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.
 
Now it's not. It's nothing but Magnitude and direction. A car going 50 MPH in a southerly direction is a vector.

A 10 1lb block slidinging down a 45 degree hill will resolve into a single vector.
Rolling Friction is a vector opposing motion and proportional to speed.
The force of Gravity (mg) acts straight down on the object.
So, there is a "resultant" force pushing on the object down the hill.

It can get quite complex, like the speed and direction of an bug walking on a rotating propeller relative to the ground. I don't even want to go there, I hated that course (Dynamics).
The first example is a physics problem.
 
In geometry a vector is the difference of two coordinate points i.e. if you subtract two coordinate points you get a vector.

Strictly mathematically speaking you are not allowed to add two coordinate points. You can add a vector to a coordinate point and the result is a coordinate point.
 
Hi

It looks 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.

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
 
Vector Space can easily be described simply as magnitude and direction or even simpler Polar co-odinates.

But "vector space" can be a lot different as well. They can be used to describe the forces on an object. An example would be a 1 kg box sitting on a 45 degree angle on a ramp. There is a force of gravity acting straight down on the box and a frictional force in the opposite direction of the incline. These can then be decomposed into a force acting in the x and a force acting in the y direction.

When stuff gets really complicated, you can talk about something like the position vector of a fly on a rotating propeller relative to the ground.

You already know about vector space with electronics with the impedance triangle.

So, what's so hard?
 
Hi,


A vector space is a mathematical concept that can be applied to many situations. It is basically a collection of vectors that are assigned to each and every point in the space. There are certain ideas about this that can be applied to many physical situations that come up, and generalities that help make using a vector space simpler than other possible methods.

A vector field is a physical concept that occurs in nature and it is described as a vector space. Here each vector at each point is simply the magnitude and direction of the field at that one single point, and the whole collection makes up the vector field.
For a 2d vector space we'd have two components for each vector in that space, for a 3d space we'd have three components for each vector in that space. There are certain properties that stand out when we do this that makes understanding what is going on simpler in many cases.
But then electronic circuits can be described as a vector space also, as can fluid flow, etc. So the vector space concept is used to describe physical phenomena just like a matrix can be used to describe physical stuff.

For an electronic example of a vector space, we might describe a two stage RC low pass filter as:
dX1=E+X1*A+X2*B
dX2=X1*C+X2*D

where
dX1 and dX2 are time derivatives of X1 and X2, and A,B,C,D are constants that depend on the circuit component values, E is the input.
 
Last edited:
Last edited:
Thank you very much, KISS, MrAl, MrT.

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

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

The links given below might also help you to understand related concepts.

3: https://www.youtube.com/watch?v=AB41vjh1JcU&feature=relmfu (linear independence)
4: https://www.youtube.com/watch?v=pMFv6liWK4M&feature=relmfu (a vector subspace and some information about vector span)

Regards
PG
 
Hi,

Here's another analogy...


Say we had a jet airplane flying due east at an altitude of 10000 feet over South Park, Colorado (he he). It's traveling at 500mph.
So we know where it is in space and how fast it is going.
Now say we had another plane just behind the first, also traveling at 500mph. This tells us that at both of those positions in space the velocity is 500 and the direction is east.
Now we take another plane and put it behind that last one, and we get another position that we now know about. The velocity there is 500 and it's also going east.
So we do that with a huge number of planes, and now we know the velocity and direction of many points in space.

But now lets take one plane and put it along side the first 300 feet away. It's going to be traveling at 490mph but still traveling east like the many others. So know we know about another position in space, where the velocity is less (490) but the direction is still the same.
Now lets put a huge number of those planes behind the first in a line. Now we know that 300 feet from the line of the first set going 500mph the velocity is 490mph and the direction is the same, east.

We also put a line of these planes on the other side of the first set too, so we see that 300 feet to the left or right of the first set the velocity is 490 but in the center line (the first set) it is 500mph.

Now we put along side those second sets of planes two more sets, each going 480mph, but still heading east.
An along side that set, we put two more sets on both sides (again 300 feet away from the previous set) that are going 470mph.

We keep doing this until we get down to 0mph where all the planes in that line are standing still in mid air.

Now we turn back to the very first plane, and this time we put another plane under that plane, and another plane over top of that plane. And this time the planes are going 500mph too. And behind both of those planes we put a huge number of planes that follow that first one. And next to those planes, we put lines of planes going 490mph, 480mph, etc., until we again reach 0mph where we have them standing still in a line.

So now we look at the entire huge set of planes in space, where we've captured their velocities at an instant of time. We see in the center line we have velocities of 500mph and as we look up or down we still see 500mph, but as we look left and right we see the velocities decreasing by 10mph every 300 feet.

What we've just seen is a flow field, in particular, a kind of laminar flow field. It's a vector field and what it tells us is the flow of the fluid at any point in space.

Note that we could change this in many ways by making some of the planes in the up and down spatial planes also decrease or increase in velocity, and that would give us a different field.

One thing though is that with this analogy the plane's motions are all captured at one instant of time, because the vectors themselves (the airplanes) dont move but remain stationary. This means that for example the very first plane we started with stays over South Park indefinitely, unlike an actual plane in the sky flying at 500mph.

So we might think of a whole collection of planes that entirely fills the space but dont actually move. This could indicate the wind speed at those points too for example. Their direction and speed tells us something about how something physical behaves in that vicinity, like the wind or some fluid.
 
Last edited:
HI again PG,

Oh you're welcome and im happy to see you are interested in these things.

If you like, you might try to map out a field if you've never done this before. This will give you a lot of insight i think.

What you could do is start with the field described by the function:
F(x,y,)=-y*i+x*j

where i is the unit vector in the direction of x and j is the unit vector in the direction of y. This is actually much easier than we might think at first.

For example, when x=1 and y=1, F(x,y)=-1i+1j, so what we do is draw an arrow that has length sqrt((-1)^2+(1)^2))=sqrt(2) and imagine that we had a new coordinate system x',y' riding on the old system so that on the x y plane where x'=-1 and y'=1 this would be an arrow pointing to the left and upward and centered at the original point (1,1).

The attachment shows this single vector. The length is sqrt(2) and the position is at the original (1,1), and because (1,1) mapped to (-1,1) we see that the new coordinate system on the right has an arrow from (0,0) to (-1,1), and then that arrow is taken over to the left and placed at (1,1). It maintains it's length and direction, but it's position comes from the original coordinate system. So that original point has the magnitude and direction as indicated by the vector.

So the steps involved here were quite simple:
1. Transform the original coordinates into the F function coordinates using the function F(x,y)=-yi+xj (this makes a new set (x',y')).
2. Draw those coordinates as a point in a new coordinate system and an arrow from (0,0) to the point, that's the vector.
3. Transfer that vector to the original coordinate system point that was used to generate it, centering the vector body on the point (this is because the vector represents that point and only that one single point so it is associated with only that one point and not really the whole arrow).

Note that the position came from the original point (x,y) but the arrow came from the new coordinate system (x',y') after (x,y) was mapped using the function F into (x',y') with the tail at (0',0').



If you go ahead and try to map some of the other vectors of that function F you should gain quite a bit of insight into the vector field concept.
For example, try doing the point (-1,-1). Note that this should lead to an arrow that is centered at the original coordinate system's point at (-1,-1).
 
Last edited:
Thanks a lot for the last post, MrAl.

Although I do have some questions about your last post, I won't ask them now. I like the way you want me (and others like me) to learn these concepts but unfortunately school doesn't want it that way. They want students to cram up a lot of material which doesn't leave any time for the 'actual' learning. Although I try to spend quite a bit of time to understand the concepts, on the other hand, honestly speaking, I don't like it personally. You can say probably I just ask these questions because I can't control my curiosity instinct. I wish I could just memorize formulae etc. and be done with it because they just want us to know the formulae and how to apply them. But whatever I'm saying is not something new because persons like Einstein also complained about it. For instance, Einstein said that the spirit of learning and creative thought were lost in strict rote learning.

I'm very much grateful to you for the help and have learned a lot from you.

Best wishes
PG
 
Last edited:
It took me a while to learn that. After I did, I got a 4.0 GPA. Goal became a "peace of paper". To achieve that goal, I managed to be a ghost in two classes that I took and since I had a co-authored published paper, I got out of English too. So, with a 4.0 GPA, I had to take a course called "College Reading and Study Skills" as my LAST course. I did wierd things like took courses in the bad school just to raise my GPA and get re-admitted and then promptly took a leave of absence. I graduated while on a leave of absence. Going to school is a process and learing is a process. Too bad they don't want them to occur at the same place or same time. Bottom line: Spit out what the teacher wants.

I ended up having arguments because I knew too much for the course, so my answers were correct when I backed them up. I's nice to get the "If you have better things to do, don't bother coming to class".

I've experienced a lot of different teaching techniques. Some of the more memorable ones were "take out a sheet of paper". Pop quizzes, sometimes 3 in a 45 minute period with instant corrections. Purpose: read the material. Then there was "Competition". In every lab class, you were given your rank in class relative to everybody else. Try "your data, their data" You had to write up your lab experiments with both your data and the data of the other groups. You don;t know anything yet: A response as to whether we were going to have a test soon. 75%/25%: Exams consisted of 4 problems, 3 were relatively easy. One was very difficult. "open/book/closed book": Thermo was done this way. The exams were part closed book and part open book. and then there is "Apply Roundup": Course designed to thin the graduating class.
Then there was the statics class with the take home exam. Talk about hard, that was hard. One hint was use a CAD program to get the dimensions of the truss.
 
Thank you for sharing this with us, KISS.

Sometimes, I get amazed at my classmates. For instance, one of them is considered to be 'extremely good' at mathematics but then I was amazed once to see that he didn't even know the basics of calculus and his concepts are worse in many other subjects. But man, he has 4.0 GPA. I'm not saying he is not brilliant. Perhaps, he is learning (or, is being forced to learn) in a way which is completely opposite of 'actual learning'. As MrAl knows some months I did a lot of hard work to have conceptual understanding of operations of operational amplifier but still my final grade wasn't good relative to other classmates who didn't have much knowledge about the operational amplifier. As I said before, I personally don't like to learn things from conceptual point of view because this is not of much use and as I can see it more clearly that your GPA is what counts at the end. Thank you.

Best wishes
PG
 
The point that I was really trying to make is that to "go to school", you have to "anticipate" the teacher's questions and answer them how he/she wants. That's the end goal. I've taken classes that way where when i red an assignment, I would make flash card of what I though the tech thought was important and put the answers on the back. Studying for the test means reviewing the flash cards every free chance you get. This worked very well for a number of courses.

When I was challenged with a list of 100 words to memorize and their looked up definitions and use them in a sentence, I went with a cassette tape played in my sleep with the word, a pause, the definition, a pause and the sentence. I reenforced it with the flash cards as well.

I don't memorize well at all and unfortunately, I didn't realize that math, Geometry, Trig, Calculus, Diff eqns. Discrete math is mostly memorization. You have to be able to do the problems with the "book closed". It took me a while to get that concept.

A medical doctor I had for a while said that he had a photographic memory and he could visualize the entire text book page. You don't realize that people are different and memory can be visual, verbal or kinestetic (repetition, how you would learn to ride a biike).

I'm mostly the latter and a conceptual thinker.

So, how would you measure the resistance of a piece of paper?

What did I leave out? What should I really be measuring?
 
Thank you.

I think you are the right person to help me with a very serious problem and that problem has a potential to disrupt my academic career quite badly. Actually, you have already stated my problem in your own words.

I don't memorize well at all and unfortunately, I didn't realize that math, Geometry, Trig, Calculus, Diff eqns. Discrete math is mostly memorization.

I'm very bad at memorizing the material myself and in technical studies you need to memorize a lot of formulae etc. When I do assignments I'm very good at solving problems because I know how to proceed logically. I have seen for many of my classmates the situation is reverse. They are good at memorizing the stuff but don't really know how to proceed logically. For instance, in the past many a time many of them were not able to do the assignment questions but I had no problem. Likewise, whenever the instructor include a question which requires you to use a little bit of logic and is different from any of the example problems, then they are in mess (I'm only talking about the guys I happen to know enough and by no means I intend to disparage them and most of them have very good grades compared to mine). But now this problem is getting more serious for me because, as you know, as you advance in technical studies the focus is more on memorization of formulae rather than conceptual understanding. What should I do? Please advise me. There is no hurry. You can do it whenever you have time. If you like you can PM me. Thank you very much.

Regards
PG
 
Last edited:
Hi,

You have to be able to memorize stuff, there's no way around it. Even in calculus I or II you have to remember a bunch of integration formulas any of which you might need on the next 'test'. So if you cant memorize for some reason, you'll have to find a way. Maybe you could take a short course on memorization or something, or look up ways that aid the memory. For example, getting more of your senses involved helps so sometimes if you read the text out loud to yourself (as funny and strange as this sounds) you'll remember it better. Then there's the technique where you find some way of remembering by remembering a phrase or something like for the color codes for resistors (there's lots of them out there) and for the musical staff notation.
 
PG: The flash card trick worked for me. The idea with Kinestetic memory is that "repetition" is the key. "sleep learning" or playing something over and over while your sleeping almost works the same way. Remember, when I did it I also re-enforced it with flash cards.

Now you have to probably switch gears 180 deg. 1. Your purpose is to get good grades. What does that mean? One, it means DO NOT waste your time trying to understand something in depth. You don't have the time for it. Be able to spit out the answer that the teacher wants. Usually, what that means is what was emphasized in class is what's important.

When you initially read an assignment, then is the time to start writing your index cards. Questions and answers on the other side or something and definitions on the other side. Now, take that deck and look at it whenever you get a chance. Watching TV, waiting for something. Possibly while eating. I did it while driving and I can't recommend that.

Mnenonics will help a lot. The trouble is being taught them: Every Good Boy Does Fine and FACE; Oscar had a Headache Over algebra; Roy.G. BIV.

What would have helped me in grade school is knowing that the digits in the 9x table add up to 9. I always had trouble with 63, 64, 54 and 56 with the multiplication tables. That simple fact would have MADE a big difference. Doing the 9x tables on your fingers is really cool too: If you want 5 * 9; put your palms out and curl your 5th finger. You have 4 fingers, the curled finger and 5 fingers or 45. Try 9 x9 and get 81. 1 x 9 = 9. Even 10*9 you get 9 and 0 or 90.

For math, you have to realize that you cannot spend time deriving stuff. There is, for instance a lot of info in 2 right triangles. the 1,2 and sqrt(3) and 1, 1 and sqrt(2) sides. So, now if someones asks you the sin of 45 deg, it's cake. 1,1 sqrt(2) has two 45 deg angles. Oscar can help you out. Sin is Oscar Had or Opposite over hypotenuse. So it's 1/sqrt(2).

You know the √2 and PI because you used it so many times. And you know the √2/2

Now, I have to show you how to do algebra my way and not the way you were taught,
1. ALWAYS use a script Y and a script X for x and y UNLESS your penmanship is EXVEPTIONAL. Caclulus is where neatness and penmanship MATTER a lot.

Take something like: 1x-3x-4y=6x+2y-10 = 0
That can be a mess, but count and reverse the sign if necessary.
e.g.
The x's: (1-3-6)x
The y's: (-4-2)y
-10=0
Note how easy it is: -8x-6y-10=0

The pages of algebra and expansions are just silly e.g. 10x(x^2+3x+4)
You know you have some X^3 and X^2 and X's in there and you can find them by inspection when other stuff is also in the equation.

You can do parenthesis that way too.

Just scan the equation for the terms, not collect them and move them around. That's just nonsense.
 
There is a lot of (good) information in this thread. I did my best to scan through it, and perhaps I missed it, but I feel one important point was not stressed sufficiently.

A key point about a vector space is that it does not require a notion of "distance" or a "norm". I feel it is important to stress this point because many of our physical examples of vector spaces are actually normed vector spaces which are a specific case of a vector space. Even when the space is not normed, often 2D and 3D spaces are shown visually as axes in space, which gives the impression that distance is an inherent property between every pair of vector in the space.

There are not many vector spaces that we use that do not have a defined norm, or at least a method to normalize the vectors in some way, but it is important to not necessarily associate the Euclidean norm (or any other particular norm, for that matter) as a necessary aspect to make it a vector space. Personally, I find it difficult to draw a 2D or 3D set of axes and not associate the Euclidean norm/distance to every vector, but that's what we need to do to understand the essence of a vector space.

What is a Normed Vector Space? Section 2 of this document helps answer this. Definition 1 says what a vector space is. Definition 2 defines what a norm is. Definition 3 defines what distance is.

https://www.electro-tech-online.com/custompdfs/2012/10/ma142blecture1.pdf
 
Last edited:
Status
Not open for further replies.

Latest threads

New Articles From Microcontroller Tips

Back
Top