Linear Algebra Linear Transformation

Welcome back to educator.com and welcome back to linear algebra, today's lesson we are going to be discussing linear transformations, so this is going to be the heart and soul of linear algebra.0000

Is this notion of a linear transformation also called the linear mapping, more often than not, I will probably refer to it as a mapping instead of a transformation, simply because it's habitual for me to think of it as a mapping, as supposed to a transformation.0012

It is true that what you are doing is you are actually transforming something but when we actually define what we mean a mapping and get into it, it will probably make more sense to call it a mapping, because we are actually taking something and literally mapping it on to something else.0028

Let's go ahead and get started, this is probably for those of you who are the physicist, engineers, and also mathematicians.0041

this is where it's going to be probably the first introduction to something that you haven't necessarily seen before, or something that you have seen before, that's going to be disgusting a very different way, a mathematical way, and in a more abstract way if you will.0053

A lot of it make scene a little odd, however just by diving in a little bit, taking a look at some of the examples, and letting them wash over you a little bit, you will realize that it is not altogether different than what you have already bee accustomed to.0071

It's just looking at it from a different angle, from a more general angle, and again that's what we do in mathematics, we take something and we try to generalize it as much as we can.0084

To take that generalization, that process of abstraction is far as we can...okay, let's go ahead and get started...0093

... This is an introduction to linear transformation and introduction to linear maps, the first thing that we want to do is generalize this notion of a function, that you have been dealing for years now.0102

We want to generalize that notion and that's what it is that we are going to be calling a map, so let's start with something that we do know, let's take the function F(X) = X².0115

Now let's talk about what this, what this means and what it is that you are actually doing here.0127

It's saying, take a number X from the real number system, do something to it, in this case square it, and then you are going to get back another number, so you are starting some place.0131

You do, you have an input, this X value, you are doing something to it, that is your function, and then you are going to end up with something else.0143

In this case if I take a 2, I end up with a 4, if I take a 3 I end up with a 9, if I take a 4, I end up with a 16, you know how to deal with this, you have been dealing with it for years now.0153

Okay, let's represent this in a slightly different way; I am going to draw a couple of pictures here, okay...0163

... Okay, and I am going to say that this is my space of real numbers, you are used to thinking of real numbers as a real number line, that's fine.0173

This is just another way of representing it as a set, as just a collection, a bag of numbers if you will, so and I call it R, because this is the real numbers.0181

And I am going to sort of duplicate that over here, now I am going to show you what it is actually going on, let me pick the couple of umbers and the real number, let's say (2, 4, and 6).0191

Here is what this function is doing, you are taking a number from the real, of this set of real numbers, you are performing an operation on it, this, so called F, which is defined by this, okay.0201

And you are getting back another number 4, you are taking a 4 and you are coming over here, you are squaring it, you are getting a 16, you are taking a 6 and you are coming over here, you are getting a 36.0216

As it turns out, what you are doing is your mapping 2 to 4, you are mapping 4 to 16 and you are mapping 6 to 36 and so on.0229

In other words, to every number in the set of real numbers, you are associating another number in the set of real numbers, so you have done, you have taken something from the real’s and you have ended up back in the real’s.0241

You can think of it as sort of ending up back in the same set, but we like to represent it this way, we like to think of them as two different sets, and we actually denote this like this.0256

We say F is a mapping, from the real numbers to the real numbers...0266

... Defined by F(X) = X², so this new symbolism is the symbolism that we are going to be using, this sort of implies that you know what's going on, but now we are sort of breaking it down, to, to say what it is that we are really doing.0276

We are taking one number, we are fiddling with it, and we are spitting out another number, something for, so we actually treat these two spaces as separate, in fact we call this the departure space...0295

... Not everyone refers to like this, but I think it is the best way to refer to it, and this is the arrival space.0313

In other words you are taking a number from the departure space, you are leaving that space, you are doing something to it, performing operating on it, whatever it is you want to call it, reforming a function.0319

And then you are arriving at some other number, some other place, the arrival space, now in this case, you are starting with a number and you are ending with a number, but that doesn't mean you always have to do that.0329

As it turns out in a minute, you will see we can start with a number and end with a vector, we can start with a vector and end with a number, or we can start even we can get even more bizarre.0339

We can start with one mathematical structure, and end up with another mathematical structure, that's why this representation is the most general, so again F is a function, is a mapping from R to R.0348

What this means is the I pick a number from the real numbers, I do something to it, and I end up in the real numbers, that's what this symbol represents, and defined by, I actually give the definition of how, of what the function is, what it is that I am doing, what operation I am performing.0363

In this case I am actually squaring a number...0378

... Let's do another example, let's say I have the function F(XY), now I have two variables, is equal to X² + Y².0383

Let's just do a simple example, if I take the point (1, 2), okay, it's X² + Y², so 1² + 2² = 1 + 4.0395

That's equal to 5, and you have probably never even thought about this before, but take a look at what's going on.0407

Now I am taking two numbers from R, and the way this is represented, notice this is actually a vector representation, when I have two numbers like this, a point in two spaces, which is what this is.0414

The point (1, 2) is also a vector, in the, in two space, so we represent that of ‘course with R₂, so when I symbolize this, according to how we did it here, here is what I am writing.0425

The function is a mapping from R₂, to R...0442

... Defined by F(XY) = X² + Y², I could also write this in vector for, the F(XY), the vector XY = X² + Y².0447

And again this coordinate XY is the same as a two vector, so what this symbolism means is that my departure space if you will is now R², it's the space of 2 vectors, my arrival space is R, the set of real numbers, I have taken a vector.0467

A vector, I have done something to the individual components of that vector, and I spit out a number, so these are two different spaces.0486

Even though I am picking numbers, there, all the numbers are from the real number line, we actually consider this thing that we take, this vector that we take as a single unit.0495

I took a two vector, I did something to the components of that two vector and I spit out a number, that's what the symbolism means, that's why this is a very powerful symbolism, and it generalizes.0507

This is a mapping from R₂ to R that means to the space of two vectors I am associating a number.0518

To the space of two, to every element in the space of two vectors, I am associating a number, I am mapping a vector to a number.0527

I am mapping a two vector to a number, that's what's going on here...0534

OK. Let's do something like this, let's define F(XYZ) =...0544

... X + Y, X+ Z, let's do an example to show what this actually looks like, if I take F(3, 2, 1), well it's just telling me X, Y, Z, this is a 3 vector.0558

In other words it is a vector in three space, if I take this, the answer that I get is well X, the first component is X + Y, so 3 + 2 is 5, and X + Z, 3 + 1 is 4.0573

I have taken a three vector, I have done something to it according to the definition of the function here, and I have spit out a two vector, so this is represented this way, pictorially.0589

This is R₃, it's the space, the collection, the set of all three vectors, (5, 6, 9z), (2, 4, 6), (1, 3, 5), (0, 0, 9), a set of three vectors, and here is my set of two vectors R₂.0601

I am taking something from here, I am doing something to it, and I am ending up in a completely different space, I am mapping it from one space to another space, this according to the symbolism is written like this.0619

F is a mapping from R₃ to R₂ defined by F of, I am going to write it in vector form.0633

X. Y, Z is equal to X + Y, X + Z, this tells me that I am taking a three vector, I am performing an operation on it, this operation specifically, and I am ending up with a two vector.0647

That's kind of extraordinary when you think about it, you have been doing it this all along, actually it's not the first time you have seen this, but to actually step back and realize that you are jumping from space to space.0664

That's pretty extraordinary and in a minute when we define what we mean by a linear mapping, it's going to be even more extraordinarily, extraordinary that you can actually retain structure, when you jump from one space to another space.0676

Okay, let's do another example...0689

... This time we will do F)XY) and we will define it as X², the second one as Y², X + Y, X² + Y².0693

In this case, let's do an example, F(2, 3), well X² is 2², that's 4, second is Y², that's 9, X + Y is 2 + 3, that's 5, x² + Y² is 9 + 4, that's 13.0710

F(2, 3) = 4, 9, 5, 13, I have taken a two vector, and I spit out a four vector, I went from two space to four space.0729

I went from R₂ to R₄...0739

... Another way of representing this is I have mapped (2, 3) to the vector (4, 9, 5, 13), just like in the previous example, I have mapped one vector to another vector, each vector belongs to a different space.1240 that's why the symbolism is actually is more general and works out beautifully, and it is a mapping from R₂ a two vector, to R₃, I know, not R₃, R₄...0743

... Defined by this, defined by F(XY) = X², Y², X + Y, X² + Y²...0783

... I am , to each vector in two space, I am associating a vector in four space, the picture looks like this, the set of all two vectors, this is R₂...0802

... Here is the set of al four vectors, R₄...0815

... That's what's happening, I treat them as separate spaces, that's what makes this beautiful, I am moving from one space to another, I can do whatever I want, something that I pull from my first space, and then I end up somewhere else.0825

It's really very extraordinary that you can do this, and it's even more extraordinary that we can actually represent the real world as operating like this...0837

Okay, so this is called a mapping, again, now what we are concerned with are linear mappings, because we are dealing with linear algebra.0847

Now we are going to actually give a definition of what we mean by linear, the examples that we gave a second ago, they are just standard mapping, as it turns out in a whole, all the possible mappings, there is a small portion of them that are actually linear.0859

And they have special properties, so now we are going to define what we mean by linear mapping, and any time we are faced with the mapping we want to check that it's linear, we are going to check this definition.0874

We will do that in a second, okay very important definition, probably the single most important, actual definition i linear algebra, definition, okay...0883

... Mapping, or a mapping, sorry, a mapping from RN to RM, let me...0899

... Make this a little bit better here...0912

... R , M, where and M could be anything, two space to six space, 1, space to 1 space, 3 space to 3 space, it's more general now...0918

... A mapping from RN to RM...0931

... Is an operation...0939

... Which to each vector...0945

... In RN, our departure space...0951

... Assigns a unique...0955

... Vector symbolized...0966

LU, in RM, don't worry about what this, it will, it will make sense in a minute when we actually start doing the examples; I'll discuss what these symbols mean.0971

Again we just want the mathematical formalism, so that our basis are covered, such that...0981

... The following hold...0991

... Okay, AL of U + V = L of U + L of v.0994

And BL of C times U = C times L of U, okay, let me just talk about what this means, real quickly and we will get the examples, so a mapping from all the space RN to the space RM.1009

Or the mapping from N space to M space is an operation, which to each vector in RN, okay associates a unique vector that is symbolized, that way...1027

... To RM, such that the following hold, notice I haven't drawn any pictures yet, I will draw pictures in a minute.1043

But it's very important to understand that these are algebraic properties, so, we have, you we are talking about a linear mapping, and yet we have drawn the lines.1049

We have drawn nothing else, this is an algebraic property, so the following has to hold, I have to check these two things, when I am presented with a mapping.1058

It says if I take, if I map U, to it's...1067

... Whatever I am associating with it, and if I map V, to whatever I am associating with it, and then I add those, it's the same as if I add them first, and then perform the operation on it, so in either order.1075

that's what this, that's what this says, that's what linear means, it means that I can either add the two elements from my departure space and then operate on it, or I can operate on it and then add them.1087

But in either case, they have to end up in the same place, and the same thing here, if I start with the vector in my departure space, multiply it by some scalar, and then operate on it.1099

meaning my function, whatever my function happens to be, it's the same as operating on the vector first, and then multiplying it by its scalar, this might seem obvious.1110

You are going to discover in a minute it's not so obvious, okay, let's draw a picture and show what this means exactly, so we have our departure space, we have our arrival space.1120

And again we are considering the spaces as collections of two vectors, three vectors whatever, so this is going to be our N, N space, our M, so they are totally different spaces, they don't have to be, but often they are.1132

Let's say I have U, and let's say I have V, and let's say I have U + V, which I can do right, if I have two vectors, I could just add them and I get another vector, because vector addition is closed.1148

I end up in the same space, over here, let's say I operate on U, and I end up with LU.1160

Let's say I operate on V, I have LV, well these are just vectors in M space, I can add vectors in M space, and I get a vector in M space.1169

This is another LU + L...1180

... Of V...1190

... Here's what linear mapping says, it says that if I take U and V, and if I do U first, and then if I operate on V separately, and then over here if I add them...1193

... I will end up here...1208

... And then if I do it the other way, if I add them first to come here, and then I operate on them,. it says I have to end up in the same place, with the same answer.1212

Think about what that means let me say that again, this is really important distinction.1224

If I have a mapping like X², well, this says that if I do X², and then if I do Y², so I, X² gives me over here, another X² gives me over here.1230

Well, if I add the X² + Y², I am going to get some number, now if I do it the other way around, if I add my X and Y first before I square them, if I take X + Y, square it, and then I do my operation.1241

If I reverse the order, it tell me that if I end up in the same place, my mapping is linear, if I don't end up in the same place, my mapping is not linear.1254

that's what linear means, it means I can do my function or my addition in either order; I still have to end up in the same place, that's what's important.1263

And we will see in a minute that that's not always the case, that linear mappings are very special things.1272

And the same thing with of ‘course the scalar multiplication, I have U, I can't end up, you know...1279

If I, I can multiply it by some scalar first, and then operate on it, or, I can operate on it and then multiply it by a scalar, I will end up in the same place, I should end up in the same place.1289

This is what I have to check every time I faced with the mapping, I have to actually put each value in there, and manually check it, you have to go through the rigorous process of actually checking to see that a mapping is linear, later of ‘course will come up with quicker ways.1299

But for the time being we want to get a feel for what linear mappings are.1314

Okay, I cannot emphasize enough how profoundly important this definition is, it is very, very, very important, this notion.1318

Spend some time with this idea, and again operate first and then add or add first then operate, in either way you have to end up in the same place.1330

If you end up in the same place, it's a linear mapping, if you don't, it's not a linear mapping.1342

Okay, examples will always clear up everything, let's do our first example which we talked about, let's say I am going to use the symbolism that I used before.1348

F is a mapping from R to R, so I am starting with a number, I am spitting out a number...1358

... Defined by F(X) = X², so you are very familiar with this function, been dealing with it for years.1367

We want to know is it linear...1374

... You might already know the answer to this, but let's actually go through the definition, and then we will talk about what linearity really means.1381

Okay, well we have to check the two, the two properties that we talked about,. we have check that vector addition is linear, and scalar multiplication is linear, that satisfies two properties from before.1387

Okay, let's do part A...1399

... We need F(X₁ + X₂), to actually be equal to F(X₁) + F(X₂), it means we need, we add the two X's first and then operate on it.1404

It has to e the same as operating on each separately, then adding them, okay that's what this mean, that's why the parenthesis are arranged the way they are, so let's check that this is the case, let's do this one first...1421

... Let's move on to a blue ink here, so I will do F(X₁ + X₂), well the definition of the function is you square it.1434

That's equal to (X₁ + X₂)², that's what this parenthesis means, anything in the parenthesis, you, that's what you do to it.1446

Well that equals X₁² + 2X₁X₂ + X₂², so that takes care of that.1455

Now let's do F(X₁) F(X₂), F(X₁) is equal to, well X₁², that’s the definition of our function right here.1468

All we were doing is we are using the definition of the function, putting in the values seeing what we get, F(X₂) = X₂².1480

We have F(X₁ + X₂), we calculated it, that's here...1493

... Here, we have F(X₁), that's here, we have F(X₂), now let's see if they are actually equal, now let's check this thing.1498

X₁² + 2X₁ X₂ + X₂², if the question is, is it equal to F(X₁), which is X₁² + F(X₂), which is X₂².1508

Are these two equal, no, they are not, this is an extra term, they are not equal, they are for.1525

We don't even have to bother checking the scalar multiplication, this is not linear.1532

It's not a linear mapping, it is a mapping from R to R, yes of ‘course it's a mapping, in fact it's a specific type of mapping called the function.1539

But again we don't want to use the word function again, we want to use the word mapping or transformation, it is a mapping, it's a perfectly valid mapping, it's a very common mapping.1549

It shows up everywhere in map and science, but it's not a linear mapping, okay...1557

... Let's try another one, let's let...1568

... Let's see, let's go back to our black ink here, let F be a mapping, again, from R to R, from the real numbers to the real numbers, meaning we start with a real number, we fiddle with it, and we get a real number back.1576

We defined by F(X) = 5X, okay.1591

we need to check, that F(X₁ + X₂, equals F(X₁) + F(X₂).1599

Okay, let's do that...1611

... F, let me actually work in red, so we go, let me go to red...1616

... F(X₁ + X₂), well, here is our definition of our function, this is what, it's 5 times the thing in parenthesis, so it's 5 times X₁ + X₂ = 5X₁ + 5X₂, distributive property.1626

Well now let's do F(X₁) = 5X₁, F(X₂) = 5X₂.1646

And now let's check to see if they are equal, we need to check that, F(X₁ + F₂ = 5X₁ + X₂.1655

The question is does it equal F(X₁) + F(X₂), 5X₁ + 5X₂, yes.1670

It checks out, the left hand side and the right hand side are the same, adding them together first, then operation on it, versus operating on each and then adding it.1679

they end up being the same, so far so good...1688

... Now we want to check scalar multiplication...1693

... F(ZX) equals, we want to check to see whether it equals, Z times F(X), well...1700

... F of...1713

... ZX...1716

... This is our 5ZX and Z times F(X), so we are checking this one, we are checking this one, equals Z times...1719

... 5X, well, 5Zx does equals Z5X, so B checks out scalar multiplication, so yes.1732

This is a linear mapping...1743

... Okay, this is a really important example, look at this where I = 5X, you know that F(X) = 5X if I write it as Y = X.1748

This is the equation of a line, that's where we get the name linear mapping, but and this is why you know that anytime you see an exponent of 1, you are talking about a line, a linear function.1759

Now you know why we call it a linear function, however it's really important to understand that linearity is an algebraic property, not a geometric property, geometry and pictures are just there to help us out.1774

We get the language that we use, for example, we call it a linear mapping, because we think of it as a line, but believe it or not, just because we can draw something, doesn't necessarily mean that there is such a thing as a line.1786

This is an algebraic property, it's a deeper more mathematical property, that has to do with mapping of moving something from one place to another, that's why it's called linear.1798

We of ‘course do it the other way, we study lines, we know that lines are equations, where the exponent on the X is 1, but this is what's going on.1808

Now you understand something very real and very deep about mathematics, this is an algebraic property, pictures are not proofs, they can help.1817

Okay., let's see what we have got, let's go back to, now let's stick with blue, okay now let's do L, call it any letter we want, as a mapping from R₃ to R₂, defined by...1829

... L of XYZ = XY, this says that if I have a 3 vector, I start with the vector, I operate on it, I get a 2 vector, and the operation that I am performing is, just take the first component, take the second component, drop the third component.1852

I am converting, transforming a 3 vector to a 2 vector, a mapping from the R₃ space to R₂ space, question is, is it linear, well let's check.1870

We have to pick a U and a V, so we will let U be the vector U₁, U₂, U₃, it's really important to keep track of all of your indices, all of your subscripts.1886

There is a lot of notational intensity going on, there is nothing particularly difficult, it's just standard arithmetic, but it's true, this does tend to get a little notationally intensive, notationally busy rather.1904

Be very careful, I mean I am going to make mistakes, believe me, equals V₁, V₂, V₃, okay, let's calculate L(U + V).1916

Well, L(U + V), well that means this is a parenthesis, so let's do U, that's L of, that means add them first, well that's just U₁ + V₁ right.1935

We are adding components U₂ + V₂, U₃ + V₃, now we can apply L to this one vector, this way.1948

That's equal to, just take the first component, U₁ + V₁...1963

... U₂ + V₂, so that's the first thing that we have...1972

... Now let's take L(U), well that's just equal to...1980

... U₁, U₂, and let's take the L(V) based on this definition, it’s just the first component, the second component, right.1992

And now let's see if these two actually equal each other, so the question is does L(U + V), does it equal, L(U) + L(V).2008

Well, L(U + V) this thing, that we got before, U₁ + V₁ U₂ + V₂.2023

Does it equal L(U), which is U₁, U₂ + this one, V₁, V₂.2037

Well when we, we can perform this addition, again we have to go as far as we can with the actual arithmetic, that equals, these are just, this is a 2 by 1 + a 2 by 1, so we add components.2051

U₁ + V₁, U₂ + V₂, the question is yes, they are, so the first part checks out.2062

We still have to check scalar multiplication, so let's go ahead and do that...2077

... Okay, now let me move on to the next page, okay.2085

We have to check that L(CU), does it equal C times L of U, well let's do L(CU), L(C) times U = L of...2091

... CU₁, CU₂, CU₃, that's equal to, we just take the first two entries, CU₁, CU₂, right.2108

And then we will calculate this, this other on, C times L(U) = C times, well the L(U) is U₁, U₂.2122

It's equal to, yeah, CU₁, CU₂, these...2134

... Mean my arrows, is you are definitely equal, so as it turns out, scalar multiplication also satisfies this equality, the definition, the second part of the definition of linearity so yes...2146

... This is a linear mapping, okay...2163

... this example that we just did, the very important mapping, let me go back to black...2173

... This mapping...2183

... Is called the projection mapping, I remember we mentioned projection a little bit earlier, or the last lesson I think, projection mapping...2188

... In this particular case, L of...2202

... XYZ is XY, this is a mapping from R₃ into R₂, all we have done is project on to the XY plane, we take it a 3 vector, you have joined a light on it, and we have just taken the projection on to the XY plane.2207

Let's draw this one more time, this is our right hand coordinate system, we have the X axis, we have the Y axis, ad we have the Z axis, let's take a vector, for example (2, 3, 5).2229

We have projected on to the XY plane, which means we drop a perpendicular onto the XY plane.2245

And this vector that we get, this two vector, because now we are only talking about two space, we are not dealing with the Z, this is that.2254

We have taken a three vector, we have projected onto the XY plane, in other words we have eliminated the Z component, and now we have this vector, which is in the XY plane, this is a two vector, this is really, and when we think about it, it's kind of extra ordinary.2266

Sometimes it seems so natural, we can actually do this, we can move from one space to another space, the space of three vectors to a space of two vectors, and not only that, we can actually retain structure, and what that means is that if you add in one space, and then operate, it's the same as operating and then adding in the second space.2280

That's what you are doing, that's amazing that you can actually do that, that structure is maintained in moving from one space to another, on the two space, it really have nothing to do with each other, they are completely different...2299

... Okay, let us , it's just terminology here, so let's go back to our picture, we have our departure space, depart, arrive.2312

Sorry about that , departure space arrival space, we are taking something from element here, we are operation it under some mapping, linear mapping in this case, and we are ending up with some other object over here, they could be two completely different spaces.2329

If we call this U, and then of ‘course we call this L(U), because we have operated on U, it's a different element now, okay, this L(U), this thing right here is called...2343

... The image...2360

... Under L, the mapping, which makes sense, you are taking this, you are doing something to it, so this is the image under the mapping, L of this original element.2364

It's the unique element associated with this, the element from the departure pace, okay, this set...2374

... Of all images...2386

... For a given set...2394

... Our given set of U's is called, you know this, seen it before, it's called the range.2397

In other words, if I have this set, let's say I only take five elements from that set, okay,(1, 2, 3, 4, 5), and let say I am only mapping those five elements, and I map over here to, let's say, let's symbolize what they look like, the images like that.2409

If I take the collection of these things in this space, that's what I call the range, the range is not the entire space, okay, it's very important to differentiate that.2428

Sometimes it can't be, sometimes we will map everything in one space, over onto another space, and it might actually end up where everything here is associated with everything here, that there are no blanks, no gaps.2439

But that's not the case, that's often not the case, the range, okay, the range is just loose things that actually end up as images under the choice is that we make from the departure space, the range is not the entire space, the range is just everything that happen to be mapped over here (2, 3, 4, 5).2451

In this case the range consists of five elements, and what we might say is, we usually this is the domain, and this is why the whole domain range thing is actually not often used, when you actually move on to, speaking about linear mappings more generally.2475

We still use the term range, we don't often really speak about the domain, so speak of the departure space, the arrival space, the individual element is an image, and the set of images for thing that you do map is called the range.2492

Okay, let's be here, what else have we got?...2509

... Okay, let's do one more example, we will let...2520

... A mapping L, be a mapping from R₃ to R₂ again, defined by...2528

... U₁, U₂, U₃, this is equal to U₁ + 1...2541

... U₂ - 3, so we are taking a three vector, mapping it to a two vector, and this is what we are doing to it, we are taking the first entry adding 1 to it, and we are taking the second and subtracting the third from it, okay/2550

An example might be L(3, 2, 2)., that's going to equal 3 + 1 is4, 2 - 2 is 0, just an example of what that mapping looks like, notice I have changed a three vector to a two vector, okay.2563

Let's check our first one, we need to check L(U + V) okay.2580

L(U + V) is equal to L of...2589

... u₁, U2, U3...2596

... + V₁, V₂, V₃ =...2601

... Let's do, we are in the parenthesis, so let's actually do it, L, U₁ + V₁, U₂ + V₂, U₃ + V₃.2609

And now I can apply L to this using this, that means I take the first entry and add 1 to it, so that's equal to U₁ + V₁ + 1, and then...2623

... U₂ + V₂ - U₃ + V₃, take the second entry, subtract the third from it, so let me put a circle around this.2637

This is what we have for that, I will go back here, and now let's calculate L(U), which is equal to first entry + 1, oops.2649

I forgot my U...2665

... This is supposed to be a U₃, to a U₁ + 1, U₂ - U₃, okay, then I will do L(V), it's equal to V₁ + 1, V₂ - V₃.2669

Now I actually check that these are equal, okay that's C...2692

... Let me move forward, I am checking that L of...2702

... Okay, now I take L(U) + L(V), let me calculate that, because that's the right side of the thing that we are going to check, that's going to be U₁ + 1, U₂ - U₃ + U + V.2709

Let me make this clear...2729

... v₁ + 1, V₂ - V₃ = U₁ + V₁...2734

... + 2, and...2747

... U₂ - U₃ + V₂ - V₃, okay.2752

When we did...2766

L(U + V), back that thing that I circled, we ended up with U₁ + V₁ + 1.2771

And U₂ - U₃ + V₂ - V₃, no I am sorry, that's not right...2781

... 2...2798

... Like I said, mistakes are easy, U₂ + V₂ - U₃ + V₃, we got this, the question is, is it equal to the thing that we just got, which is L(U) + L(V).2802

Well, is it equal to U₁ + V₁ + 2 and U₂, U₂ + V₂.2822

I am going to rewrite this, so that it actually looks -U₃ + V₃, so notice...2835

... This matches that, that's fine, however this is not the same as this, when we wrote it all out, we ended up with two things that are not equal, so this particular mapping...2847

... is not linear, so when we are given a mapping, we have to use examples, general examples, U and V, we have to check that, adding these two vectors, then operating on it, is equal to operating on those two vectors separately and then adding them.2864

If those two are equal, that's half of it, then we have to check to see whether, take us just any old vector, multiply it by a scalar, then operate on it.2884

If it's the same as operating on the vector, and then multiplying it by a scalar, if it ends up in the same place, the mapping is linear.2893

That's the definition of a linear mapping, it has to satisfy those two properties, that operation and addition, operation and multiplication, you could reverse the order.2901

You still end up with the same place in you arrival space, if you end up in different places, the mapping is not linear, [profoundly important, profoundly important definition, and we are going to spend the rest of this, the rest of the semester, the rest of all the lessons, discussing everything that this ultimately implies.2910

Thank you for joining us today at educator.com for linear algebra, we will see you next time.2929