A more formal understanding of functions | Matrix transformations | Linear Algebra | Khan Academy

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I think you have touched upon the idea of \u200b\u200bpairing to Somewhat in mathematics But what I want to do in this presentation is to clarify it a little bit Deeper than what you are accustomed to, then I will link it to some vector terminology Linear algebra that we have seen previously . So in fact pairing is a relationship between elements One group and other group elements Let’s say we have elements of group x, and each element of this group x I will link this item With another element of group y Group y So if you assume this is group x, and this is group y– And y doesn’t have to be smaller than that, that way I drew it – the pairing is a relationship Since if I take an element of group x, let me see this The item I will take We’ll consider it a point This conjugation is guaranteed, if you give me an element of group x, then I will I give you an element of group y that is associated with That element of group x So the pairing content, give me this, and I plug it in With that element This means linking it With another element of group y And if you would give me another point here I would link it to another element of group y I will link it to another element of group y And I can bind it to the same element in group y This term is guaranteed to bind an element of the group x, I speak very general terms With another element of group y And based on that you will say, well this Very brief, so how does it have to do with the conjugations that are Have we seen it in the past? Let me write a link here that you may have seen For many times in the past. You dealt with (f (x = x ^ 2 How can we write this with this phrase? Well this is a conjugation, suppose this is The traditional way you see it This pairing Let me write it with f, as I used to write it in a picture (g (x), not always written with f, but I think you got the idea In this case, f is one of the real numbers – the real numbers It’s everything that can be put here – actually This is part of the definition of pairing I can restrict it to be limited to only integers, or even even numbers Or be correct marital numbers But this is part of the definition of conjugation, that I am I define the pairing to be among the real numbers What I am saying is that you can put any real number here, and It will be among the real numbers In this case, if x is a real number, it will correlate By himself, this is very logical If this is the real set of numbers – and clearly The real numbers will go in all directions indefinitely – however If this is for real numbers, connect this pairing will be by Take each point in R and connect it with Another point from R That is, it is about taking each point and linking it to a relationship Full square And I want to be a precise concept – or at least On my mind when I touched on the associations for the first time I was thinking, if you give me x and square it, and I give you the x squared And it’s true, this is what you do Or at least the way it worked my mind, meaning what I thought Is to change the x to another number And maybe you can think of it this way, and maybe it is that The best way to deal with it But the mathematical definition you started with Here it was more than just connecting x with x ^ 2 This is another concept of conjugation by writing The exact same thing These two phrases here, this phrase and that The phrase is identical Perhaps you have not seen this phrase before, but I prefer it It explains the link Better while this binding I would imagine as if you were putting x In a meat grinder or in any other machine It will give you the x or the x squared, or do the necessary With the x For me, this term means actual connection If you give me x, I will link it Another real number is called x ^ 2 That is, we will have another point This is by using some terms, I guess You have seen these terms before, the group that You link it to is the domain it is part of Definition of pairing I, as a conjugation, is guaranteed that all An entry fit here must belong to the real set of numbers Now the group you linked to is called the corresponding field Corresponding field The straightforward question you will probably ask now When I learned all this about pairing in algebra 2 For the first time, we did not use The term corresponding field We did not use the term corresponding field We have this idea of \u200b\u200ba domain, and I learned the term domain when I was in the ninth or tenth grade How does this corresponding domain relate to the domain? It is a very accurate concept So the corresponding field is a group to which it is related, and in this Example This would be the corresponding field In this example, the real numbers are Domain and corresponding field The question now is how does the domain relate to this? So the corresponding field is the group that can Link to it It is not necessarily related to all Point in the corresponding field What I am saying is that this pairing generally links Items from this group with that group Scope is a subset – let me write this way It can be equal to the corresponding field Any subgroup It is a subset of the same group, each element of the group It is an element in itself, that is, it is a subset of itself So the range is a subset of the corresponding field where the conjugation is Associated with it As the pairing is associated with it Let me give you an example Suppose I know the conjugation g, which is about A link from a set of real numbers Let me say that he got from R2 to R That is, I am taking two rows Connect them with R I’ll define g, and write it down In a different way Now I’m going to take two values \u200b\u200bfor g, and I can say xy or I could say x1, x2 Let me do it this way g (x1, x2) = 2 It’s a correlation from R2 to R, but it’s always equal to 2 And let me write the other concept of it You may not have seen it before I can write g g connects any point x1 and x2 to point 2 This makes the connection a little simpler But until you get the idea right, what field do we have? What is our field? What are the real numbers? This was part of the definition of conjugation that I have, as I have said We connect from R2, so our range is R2 . Now what is our corresponding field? Corresponding field Our corresponding field is the group to which I associate It is part of the definition of conjugation By definition, this is the opposite field So the corresponding field is R Now what is our pairing range? What is our pairing range? A range is a set of values \u200b\u200bthat Associated with pairing In this case, we will always associate with the value 2, so the range is The value is only 2 And if we want to look more closely at this – actually R2 – I do not want to paint it with a costly advertisement, but rather I will draw it With the image of my Cartesian axis, but I will give you a brief concept This R2 And if I had to draw R, I would draw it with a picture Preparation line In fact, let me draw in this way just for fun, not It is normal to see it written this way But I can draw R like this R2, and I can Draw R as a straight line So this is group R I can draw it that way, but let’s Assume that this group R And the conjugation g will connect any point here Point 2 2 represents a small point of R The conjugation g takes any point in R2, i.e. the Cartesian coordinate system, this Point 3, -5, that is, point It will connect to point 2 in R If you think the point that connects it to point 2 This is what g always does So the corresponding g field – you can say it is all Real numbers, but only 2 Now if I wrote the example – then I assumed that Let me solve another example, and it might be very interesting If you write that h is an association that passes from R2 to R3, and I’ll be precise here, h It passes from R2 to R3 And I’ll write here (h) (x1, x2 = – so now I’m connecting space after a top, so I’m going to assume This is equal to, say first The axis or first component of R3 is x1 + x2 Suppose the second axis is x2 – x1 And let’s suppose the third axis is x2 x x1 Now what is our domain, domain and corresponding field? So the field by definition is this The corresponding field by definition is R3 And notice that I go from space with two dimensions To another space consisting of three dimensions, or three Vehicles But I can always connect a point with x1, x2 with Some point in r3 And here’s a little elusive question, what is the range? Can I always connect each point – maybe not The best example is because it is not simple – but can I Connect each point in R3 – so this is the opposite field Our field is R2, and the conjugation goes from R2 to R3, then h So the range, you see, is not the same as any Axis you can express in this way Let me give you an example For example I can put x1 and x2 and find the output Let’s do this Let’s take h – let me use the other concept – let me Suppose I say h, and I want to find the link From the point in R2, suppose it’s 2,3 Then the pairing tells us that this is related With the point in R3 You add the two phrases, 2 + 3, equal 5 And I’ll find the difference between x2 and x1 – so 3 – 2 = 1– and then multiply the two, 6 So clearly, this will be in the range, it is Element of the range I’m going to write it like this: it’s an element in range So for example point 2.3, and you will probably be here It will connect to the three-dimensional point, it Graphic advertisements commissioned here But I think you understand the idea, it will relate With a three-dimensional point, 5,1,6 So undoubtedly this is an element of scope My question to you now, if I have some point on R3, let’s We assume this is the point 5,1,6 Is this point an element of the range? It undoubtedly has an element of the corresponding field, it lies on R3 She is undoubtedly here, and this according to Definition is the corresponding field But is she in range? Well, if you take The sum of the two numbers must be 5, and 1 is The difference between the two numbers, and 0 is Output of the two numbers We know very clearly that 5 is the sum, and 1 is the difference We deal with 2 and 3, and there is no way we can get there The result of these two numbers is 0 So this is not a range So the range is going to be a subset of all of these points On R3, then there will be a ton of points not included Range, and there will be a subset smaller than R3 Out of range Now I want to introduce you to one of the terms That are related to functions These pairings are at the top, this pairing In points in R2 to R, then the corresponding field is R This pairing above is perhaps the most common In mathematics, which is also Associated with R These functions that are associated with R are called the standard value or The real value, depending on how you think about it But if they relate to one space dimension, we call them Standard value associations, or real value associations It is all of these functions that I dealt with it on top to the point on The field of mathematics, if you have not taken radial differentiation Now the functions that are associated with spaces or subspaces that contain On more than one dimension – if you relate to R or any subset From R, you would have a real value pair, or Coupling a record value If you relate to Rn, where n> 1, if it correlates With R2, R3, R4, R100, so you’re dealing with Associate a vector value So this last conjugation that I defined here, h is Associate a vector value Anyway I think you have at least now Mathematical concepts tools that help you understand what I am doing in The rest of the shows, and I hope you found this Useful

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