Linear Algebra Vector Spaces

Welcome back to educator.com and welcome back to linear algebra.0000

Last time we discussed lines and planes. Today we are going to move on to discuss the actual structure of something called a vector space.0004

So, for those of you who come from the sciences and the engineering, physics, chemistry, engineering disciplines... this notion of a vector space may actually seem a little bit strange to you.0014

But, really, all we are doing is we are taking the properties that you know of, and are familiar with, with respect to 2 space and 3 space, the world that we live in, and we are abstracting them.0027

In other words we are looking for those properties of that space that are absolutely unchangeable. That are very, very characteristic of a particular space and seeing if we can actually apply it to spaces of objects that have mainly nothing to do with what you know of as points or vectors in this 3-dimensional space.0037

As it turns out, we actually can define something like that, and it is a very, very powerful thing.0058

So, again, when a mathematician talks about a space, he is not necessarily talking about n-space, or 3-space or 2-space, he is talking about a collection of objects that satisfies a certain property.0064

We give specific names to that particular space, for example, today we are going to define the notion of the vector space.0076

Those of you that go on into mathematics might discuss something called the Bonnock space, or a Hilbert space.0081

These are spaces, collections of objects that satisfy certain properties.0087

Before we actually launch into the definition of a vector space and what that means, let us recall a little bit what we did with linear mappings.0094

So, you remember your experience has been with lines and planes and then we introduced this notion of a linear mapping.0103

We told you this idea of a linear mapping has nothing to do with a line, we are just using our experience with lines as a linguistic tool to sort of... the terminology that we use comes from our intuition and experience.0110

But, a linear mapping has nothing actually do to with a line. It has to do with algebraic properties.0123

In a minute we are going to be defining this thing called a vector space, which is a collection of objects that satisfies certain properties.0129

As it turns out, even though we call it a vector space, it may or may not have anything to do with vectors. Directed line segments, or points in space.0137

Granted, most of the time we will be working with our n-space, so actually we will be talking about what you know of as vectors or points, but we are also going to be talking about say, the set of matrices.0146

The set of say 5 by 6 matrices. It has nothing to do with points and the certainly do not look like vectors. They are matrices, they are not directed line segments.0159

But, we use the terminology of vectors and points because that is our intuition. That is our experience.0167

In some sense we are working backwards. We are using our intuition and experience to delve deeper into something, but the terminology that we use to define that something deeper actually has to still do with our more superficial experience about things.0174

I do not know if that helped or not, but I just wanted to prepare you for what is coming. This is a very, very beautiful, beautiful aspect of mathematics.0190

Here is where you sort of cross the threshold from -- we will still be doing computation, but now we are not going to be doing computation strictly for the sake of computation.0198

We are doing it in order to understand deeper properties of the space in which we happen to be working.0207

This is where mathematics becomes real. Okay.0214

Okay. Let us start off with some definitions of the vector space.0220

Now, these definitions are formal and there is a lot of symbolism and terminology. I apologize for that. We will try to mitigate that as much as possible.0224

A lot of what is going to happen here is going to be symbols, writing, and a lot of discussion.0233

We want to get you to sort of start to think about things in a slightly different way, but still using what you know of, regarding your intuition.0237

Not relying on your intuition, because again, intuition will often lead you astray in mathematics. You have to trust the mathematics.0247

Okay. So, let us define a vector space. A vector space is a set of elements with 2 operations.0254

Now, I am going to have different symbols for the operations. The symbols are going to be similar to what you have seen as far as addition and multiplication, but understand that these operations do not have to be addition and multiplication.0275

They can be anything that I want them to be, plus, with a circle, and a little dot with a circle, which satisfy the following properties.0285

Okay. There are quite a few properties. I am going to warn you.0314

We will start with number 1. If u and v are in v... a set of elements, let us actually give it a name instead of elements... v.0320

Then, u + v is in v. This is called closure.0343

If I have a space and I take two elements in that space, okay? Oops -- my little symbol.0350

If I, in this case -- that is fine -- we can go ahead and call this addition, and we can call this multiplication as long as we know that this does not necessarily mean addition and multiplication the way we are used to as far as the real numbers are concerned.0355

It could be any other thing, and again, we are just using language differently. That is all we are doing.0370

So, if u and v happen to be in v, then if I add them, or if I perform this addition operation on those 2 elements that I still end up back in my set.0378

Remember what we did when we added two even numbers? If you add two even number you end up with an even number.0390

In other words, you come back to the set... but if you add to odd numbers, you do not end up back in the odd number set, you end up back in the even number set.0394

So, the odd numbers do not satisfy the closure property. That means you can take two elements of them, add them, but you end up in a different set all together. That is very odd.0403

That is why we specify this property. So, if you take two elements of this space, the vector space, then when you add them together, you stay in that vector space, you do not land someplace else.0410

Okay. The others are things that you are familiar with.0422

u + v = v + u, this is the commutativity property.0427

By the way, I am often not going to write this... these circles around it. I will often just symbolize it like that and that.0440

Again, they do not necessarily mean addition and multiplication, they are just symbols for some operation that I do to two elements.0448

Okay... B. u + v + w = u + v + w... associativity.0457

This is the associativity of addition operation.0480

C. There exists an element, a symbolized 0-vector in v such that u + 0, u + that element... excuse me... is equal to 0 + that element... commutativity... = 0.0486

This is just the additive identity. There is some element in this vector space that when I add it to any other vector in that vector space, I get back the vector, nothing changes.0513

Okay... and d, for each u in the vector space, for each vector in the vector space, there exists an element symbolized -u, such that u + this -u = that 0 vector element.0524

This is called the additive inverse, 5... -5... 10... -10... sqrt(2)... -sqrt(2).0552

This says if I have any vector, pick any vector in a vector space, in order for it to actually satisfy the vectors of the vector space, somewhere in that vector space there has to be an element, the opposite of which when I add those two together, I end up with a 0 vector.0558

That is what it is saying. Okay.0572

2... so this first one is the set of properties having to do with this addition operation.0575

Number 2 happens to deal with scalar multiplication operation.0580

If u is in the vector space v, and c is some real number, scalar, again, then c × u is in 5.0589

Again, this is closure with respect to this operation.0603

Okay. We did closure up here. We said that if we do this addition operation, we still end up back in the set.0609

Well, this one says that if I multiply some vector in this space by a number, I need to end up back in that set... I cannot jump someplace else.0614

This is closure with respect to that operation. That one was closure with respect to that operation.0625

Okay. I am going to... I am not going to do (a,b,c,d) again, I am going to continue on (a,b,c,d,e)... c × u + v, and again, you have seen a lot of these before... c × u + c × v... this says that it has to satisfy the property of distribution.0631

c + d × u = c × u + d × u... distribution the other way.0654

The distribution of the vector over 2 scalar numbers.0663

G is c × d × u... I can do it in any order.0669

c... d... × u.0682

H, I have 1, the number one × u is equal to u.0688

Okay. Let us recall again what this means. If I have a set of elements that have 2 operations, 2 separate operations, two different things that I can do to those elements.0698

Either I can take two of them, and I can add them, or two of them and multiply them.0709

They have to satisfy these properties. When I add two elements, they still up at the set, they commute.0714

I can add them in any order. There exists some 0 in that set such that when I add it to any vector I get the vector back.0720

And, there exists... if for every vector u... there exists its opposite so to speak, its additive inverse.0728

So, when I subject it to addition of those two, I end up with a 0 vector, and scalar multiplication.0735

Closure under scalar multiplication, it has to satisfy the distributive properties and what you might consider sort of an associative property here.0742

And, that when I multiply any vector here × the number 1, I end up getting that vector again.0752

Set two operations. They have to satisfy all of these. I have to check each and every one of these. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.0758

Now, it is true. You do have to check every one of these, but later we will develop... very shortly we will develop some theorems that will allow us to bypass a lot of these and just check 1 or 2 to check to see if a particular space satisfies these properties.0767

When we do the examples in a minute, we are not going to go through excruciating detail.0781

I imagine some of your homework assignments require you to do that, and that is part of the process, is going through this excruciating detail of proving... of making sure that every one of these properties matches or is satisfied.0787

Going through that process is an important process of wrapping your mind around this concept of a vector space, because it is precisely as you go through the process that you discover surprises...0801

That what you thought was a vector space actually is not a vector space at all. Do not trust your intuition, trust the math.0812

Many, many years and hours of labor have gone into these particular definitions. This is a distillation of many year of experiences, hundreds of years of experience with mathematical structures.0818

We did not just pull them out of thin air. They did not just drop out of the sky. These are very carefully designed. Very, very specific.0828

Okay. Let us move forward.0838

Now, the elements in this so-called vector space, we call them vectors.0842

But, they have nothing to do with arrows. They can be any object, we are just using the language of a n-space, 3-space, 2-space, 4-space to describe these things.0849

We call them vectors, but they do not have any... they may not have anything to actually do with directed line segments or points.0860

Okay. Let us see... vector addition, scalar multiplication... okay.0869

When the constant c is actually a member of the real numbers, it is called a real vector space.0877

We will limit our discussion to real vector spaces.0885

However, if those constants are in the complex numbers, it is called a complex vector space.0890

Complex vector spaces are very, very important. As it turns out, many of the theorems for real vector spaces carry over beautifully for complex vector spaces, but not entirely all of them.0906

Again, this and this are just symbols. They are abstractions of the addition and multiplication properties.0918

When we speak about a specific vector space, like for example the vector space of real numbers, then addition and multiplication mean exactly what you think they are.0930

But we need a symbol to represent the operations in other spaces. These are the symbols that we choose, because these are the symbols that our experience has allowed us to work with.0943

Okay. Let us just do some examples. That is about the best way to treat this.0957

Okay. Let us consider, so our first example, let us consider RN, n-space... with that and that, meaning exactly what they mean in n-space.0968

The addition of vectors, multiplication by... multiplication by a scalar.0990

As it turns out, RN is a vector space.0998

Now, again, we are not going to go through and check each one of those properties. That is actually going to be part of your homework assignment.1002

For a couple of these in a minute, we will be checking a few of them, just to sort of show you how to go about doing it.1009

You are going to go through and check them just as you normally would, so it is a vector space.1016

Well, for example, if you wanted to check the closure property, here is how you would do it.1022

Let us take -- excuse me -- let us deal with a specific one, R3, so you let u = u1, u2, u3... and you let v = v1, v2, v3, then you want to check the closure of addition.1027

So, you do this, you write u, that little symbol is u is equal to u1, u2, u3, and in n-space, that symbol is defined as normal addition of vectors.1047

Remember, we are adding vectors, we are not adding numbers, so this is still just a symbol... v1 + v2 -- I am sorry, not plus.1065

v1, v2, v3, well, that is equal to... we are going to write it in vector form... u1 + v1, u2 + v2, u3 + v3... well, these are just numbers.1074

So, you end up with a number, a number, a number. Well, that is it. You just end up with a number, a number, a number, and that definitely belongs to R3. This is a vector in R3.1093

We started off with vectors in R3, you added them, you ended up with a vector in R3, so closure is satisfied. That is one property that you just checked.1105

You break it up into its components, you actually check that these things matter... Okay.1114

Alright, let us check this one. Let us see. This one we will do in some detail.1123

Consider the set of all ordered triples, (x,y,z), so something from R3, but we are only taking a part of that.1133

And... define this addition operation as... so (x,y,z) + (r,s,t) = x + r, y + s, z + t.1157

So, this addition operation is defined the same way we did for regular vectors. No difference there.1181

However, we are going to define this multiplication operation as possible this way. We are going to say that c ×... use my symbols here, it is the point we are trying to make... (x,y,z) = c(x,y,z).1188

Now, I am defining this multiplication differently. I am saying that when I have c × a vector in R3, that I only multiply the first component by c, I do not multiply the second and the third.1212

I can define it any way that I want. Now, we want to check to see that under these operations, is this a vector space? Well, let us see.1228

As it turns out, if you check most of them, they do. However, let us check that property f.1238

So, we want to check the following... property f was c + d × u... does it equal c × u + d × u.1244

So, we want to check this property for these vectors under these operations.1265

Notice, this particular property relates this operation with this operation as some kind of distribution. So, let us see if this actually works.1271

Alright. Now, once again, we will let u, component form, u1, u2, u3, well let us check the left side.1280

So, c + d × u is equal to, well, we go back to our definition, how is it defined... it says you multiply the thing on the left side of this symbol only by the first component.1296

So, it is equal to c + d × u1, but the second and third components stay the same. I hope that makes sense.1318

Our definition is this multiplication gives this. That is what we have done. c + d, quantity, this symbol × u.1330

Well c + d quantity × that. Okay.1337

It is equal to cu1 + du1, u2, u3.1342

So, we will leave that one there for a second.1357

Now, we want to check cu + du, so that is that... now we are checking this one here.1362

cu + du, well, c... should put my symbols here, I apologize. Let me erase this.1380

c · u + d · u, okay.1400

So, c · u, again we go back to our definition,1407

That gives me cu1, u2, u3 + du1, u2, u3 = cu1 + du1, u2 + u2, because now this symbol is just regular addition, u3 + u3 equals...1411

Now, watch this. cu1 + du1, well u2 + u2 is 2u2, u3 + u3 is 2u3.1457

Okay. That is not the same as that.1473

This is cu1 + du1, yeah the first components check out, but I have u2 and u3, here.1481

I have 2u2 and 2u3 here. That is not a vector space.1487

So, if I take the set of ordered triples, vectors in R3, and if I define my operations this way, it does not form a vector space.1494

You are probably wondering why I would go ahead and define something this way.1507

As it turns out, I am just going to make a general comment here. The real world and how we define things like distance, they are defined in a specific way to jive with the real world.1511

There is no reason for defining them that way. In other words there is no underlying reality or truth to how we define things mathematically.1527

What is important is the underlying mathematical structure and the relationships among these things.1535

This is why something like this might seem kind of voodoo, like I have pulled it out of nowhere.1540

I have not pulled it out of nowhere. As it turns out, you run across things like this. In this particular case, we know how vector spaces behave.1548

Well, if I can check this one property, having come up with a new mathematical object... let us say I happen to have to deal with something like this and I discover it is not a vector space.1555

That means that everything that I know about a vector space does not hold here. So, I can throw that thing out. I do not have to go and develop an entire new theory for each new space that I work with, that is why we do what we do.1564

Okay. Another very, very important example. Let us consider the set of m by n matrices.1579

So, if I take the set of 2 by 3 matrices... all the 2 by 3 matrices of all the possible entries, there is an infinite number of them.1595

The matrix by itself is an element of that set. The question is, is it a vector space?1603

We define, of course we have to always define the operations, we define the addition of matrices as normal matrix addition.1613

We define the scalar multiplication thing as normal scalar multiplication with respect to matrices which we have done before.1630

As it turns out, if you check those properties, yes, the set of m by n matrices, which is symbolized like this, is a vector space.1644

So, once we have checked the properties, and that will definitely part of one of the homework assignment, I guarantee you that that is one of the assignments you have been given.1660

IT is just something that we all have to do, is check for matrices that all of these properties hold.1667

Now, when I speak about a set of let us say 2 by 2 matrices, I speak of any random matrix as a vector.1674

Because, again, it satisfies the certain properties and I am using the language of n-space to talk about a collection of objects that has nothing to do with a point.1684

A vector is not a point. As you know it from your experience, but, I can call it a point in that space. In that collection of 2 by 2 matrices.1691

That is the power of the abstract approach. Okay. Let us see... here is an interesting example.1704

Let us let v be the set of all real valued functions. Real valued functions just mean the functions that give you a real value, x², 3x, 5x + 2, sqrt(x), x³, something like that.1715

... functions f(t) defined on a given interval. We will actually just pick an interval, we do not have to, this is true when it is defined on all the real numbers, but we will just choose this particular interval a, b.1739

We define our addition as follows... f + g(t) = f(t) + g(t), and c not f, the symbolism is going to be kind of strange, I will talk about this in a minute... c × f(t).1759

Okay. I am taking my set of all re-valued functions. Just, this big bag of functions and I am saying that I can pull... I can treat that as a space.1792

I can define the addition of two of those functions the way I would normally add functions.1802

I can define this scalar multiplication the way I would normally define scalar multiplication... just c × that function.1808

Notice the symbolism for it. Here on the left, I have f + g(t).1816

In other words, I am somehow combing two functions in a way that, again, the symbolism is a little unusual and you probably are not used to seeing it like this.1823

But these are the actual definitions of what it is that I have to do when I am face with two functions, or if I am multiplying a function by a scalar in this particular space.1834

Now, as it turns out, this is a vector space. What that means is that the functions that you know of x², 3x², you can consider these as points in a function space.1845

We call them vectors. They behave the same way that matrices do. They behave the same way that actual vectors do.1862

This is what is amazing. There is no reason to believe that a matrix, or a function, or a point should behave the same way.1870

As it turns out, their underlying algebraic properties are the same. I can just treat a function as if it were a point.1878

Okay. Let us see. In general, when you are showing that a given space is a vector space, what you want to do is you want to check properties -- excuse me -- 1 and 2.1890

You want to check closure for the addition property, and you want to check closure for the multiplication, for the scalar multiplication operation.1914

Then, if those are okay, if they are not... then no worries, you can stop right there, you do not have to worry it is a vector space.1921

If those are okay... then you want to check property c next.1929

Okay. A little bit of a notational thing, from now on, when we write u + v, just to save ourselves some writing, we are just going to write it as normal addition.1939

That does not mean that it is normal addition, it just means that this is a symbol describing that operation.1956

We just have to keep in mind what space we are dealing with, and what operation we are dealing with.1963

Let us see... let us talk about some properties of spaces.1970

So, if I am given a vector space, these are some properties that are satisfied.1977

0 × u = 0 vector. Notice this 0 is a vector, this 0 is a number.1984

This says that if I take the number 0, multiply it by a vector, I get the 0 vector.1993

So, they are different. They are symbolized differently. b... c... × the 0 vector is the 0 vector. c is just a constant.1998

If cu = 0, if I take a vector and I multiply it by some scalar and I end up with a 0 vector, then I can say that either c is 0, or u is 0.2018

Either the scalar was 0 or the vector itself was the 0 vector.2050

d - 1 × the vector u will give me the additive inverse of that vector, -u.2056

Okay. Now, what we are going to do is we are going to show in detail that the set of all real valued functions, what we did before, for all real numbers is actually a vector space.2068

Okay. Let me see here. Yes. Okay, let us go ahead and check property one which is closure.2086

So, again, we are talking about the set of all real valued functions.2102

Remember our definitions? We defined f... actually we are not using that symbol anymore, we are just going to write it this way.2107

We said that f + g(t) = f(t) + g(t). Okay.2116

So, we want to check closure.2124

Does it equal f(t) + g(t), that is not a question, that is our definition... the question is, is that a member of s, which is the set of real valued functions.2135

If I take a function, and I add another function to it, I still end up with another function.2157

So, yes, closure is satisfied. If I add two functions, I get another real valued function, so closure is satisfied, it stays in the set.2161

2. If I multiply, I know that c... f(t), the definition was c × f(t)... well, if I take some function and I multiply it by a scalar, like if I have x² and I multiply it by 5x², it is still a real valued function.2172

So yes, it stays in the set. So closure it satisfied for scalar multiplication. Property 2.2191

Okay. Let us check property c. In other words the existence of a 0.2200

So, property c, in other words, does there exist a function g(t) such that f(t) + g(t) gives me back my f(t).2208

Well, yes, as it turns out, g(t), the 0 function, I will put f for function, there is a function which is the 0 function, it is a real valued function.2238

So, for example, 0 f(3), well, it gives me 0. It is a function... if I use 3 as an argument, it gives me the real number 0.2252

It actually exists, it is part of that space, so it does exist, so yes this property, so there is such a thing and it is called the 0 function.2264

Okay. Let us check property d in a little bit more detail. Let us see.2275

So, we want to check the existence of an inverse. So, f(t), we want to know if something like that exists.2285

Well, now the question is... if I have some given function and I just take the negative of that function, is that a real valued function?2303

Well, yes, it is. If I have the function x², if I take -x², it is a perfectly valid, real-valued function and it is still in the set of real valued functions.2316

So, yes, there is your answer.2327

That additive inverse actually exists, and it is in this set, so that property is satisfied.2331

Let us check property f. We want to check whether c + d · f(t) = c · f(t) + d · f(t).2340

Well, this left hand side is going to be this one, c + d · f(t), is defined as c + d f(t) = c × f(t) + d × f(t).2368

Now, this part, I will bring down here, c f(t) = c f(t), d · f(t) = d f(t), and of course, when I... this is this... this is this.2401

When I add them together, normal addition, normal addition... I end up with exactly this.2439

So, it turns out that when I add these two do end up being equal, so yes, once again, the set of functions, all real valued functions on a given interval or in this case over the real numbers, they do form a vector space.2447

So, again, that is actually pretty extraordinary when you think about it, that functions behave the same way that points in space do.2468

For those of you that actually continue on into mathematics, it is going to be a really, really interesting thing when you can actually define the distance between two functions.2474

When I say distance, again, we are talking about distance, but as it turns out the notion of a distance in a function space is actually entirely analogous to the distance between 2 points in say 3-space.2486

Because again, 3-space, and the function space are both vector spaces.2500

They satisfy a certain basic kind of properties, and that is what we are doing when we define a vector space. We are looking for deeper properties, fundamental properties that are satisfied for any collection of objects.2506

That is why it is so amazing that there ultimately is no difference between this base of functions, real space, R2, the space of 2 by 3 matrices, their algebraic behavior is the same, and that is what makes this important, that is what makes this powerful.2518

Thank you for joining us here at Educator.com, we will see you next time. Bye-bye.2536