Linear Transformations , Example 1, Part 1 of 2

all right in this video we're going to start talking about linear transformations and in this context in general a transformation is just going to be an operation that takes a vector and maps it to another vector doesn't have to be in the same space at all as you'll see so I'll give a kind of a simple example and then we'll talk about justifying whether or not some given transformations are linear or not so again a little definition we say so to be a linear transformation the following satisfied so it says if T is a linear transformation we have two vectors U and V sees a scalar it says the following hold it says if we add our two vectors together and then apply the transformation it says we get the same thing as if we if we were to take that the transformations individually and then add them together it says that will always be equal the second condition says if we take our vector multiply it by a scalar apply the transformation we'll get the same thing as if we were to take the transformation and then multiplied by the scalar ok so again what I just said there it is in words I'm not going to read all those it's what I just said so feel free to take a look at that again if you want to think about it there for a second so let's look at a couple examples here we're going to determine whether or not these transformations are linear or not okay so notice in Part A we've actually got a vector we take a transformation from a vector in r2 and it actually Maps it into r3 Part B we've got a vector that starts in r2 and again stays in r2 but you can kind of create any old rule that you want and again you know that vectors can go wherever you wanted to go okay and again you know a transformation doesn't have to be linear and that's what we want to figure out so again just to show you an example of a transformation this is not the proof at all yet you know maybe we look at this first this first example you know if we started with the vector for you know 15 so that's a vector in r2 it says under this this mapping all it says is take the first component subtract away the second component so we would have 4 - 15 then it says add the two components together 4 + 15 and then it says take the first component and multiply it by 2 so we would get 2 times the first component which is 4 and well 4 minus 15 what is that negative 11 4 + 15 that's 19 + 2 times 4 that's going to be 8 so under this transformation it's going to map the vector with components for 15 into r3 and it's going to give us the vector with components negative 11 19 and 8 so again nothing crazy there that's all it is just taking one vector has some rule associated with it and it produces a new vector alright let's decide if this transformation is linear or not and oftentimes again when you're trying to prove something or justify something it's just a matter of using the definitions so I'm going to break this one up into two videos and part one will justify this first condition so just T of U plus V does that equal the transformations individually added together so obviously this first condition doesn't hold there's no point in showing the the you know the second condition about the scalar multiplication because you know obviously if one condition isn't old it's not linear but if this one holds we'll go back and show that the scalar multiplication deser doesn't hold okay so the way I'm going to do this is I'm just going to pick some vectors I'm going to let vector u have components a1 a2 will let vector V have components let's say b1 and b2 and now I'm just going to start applying things individually so if we take T of U plus V so I'm going to find an expression for this left side I'm going to get an expression for the left side then I'm going to come back figure out an expression for the right side and we'll just compare them see whether or not they're equal so vector U that's a 1 and a 2 Plus vector V which has components b1 b2 okay so again we know how to add vectors together we just add them a component a component at a time so we would get a1 plus b1 and then we would get a2 plus b2 alrighty now I'm going to go back to my actual rule here and this is the rule this tells us what happens in this transformation so again you know take the first component - the second add them together and then multiply the first component by 2 so if we apply that transformation it says again the first thing that happens we said was we take a1 plus b1 we take the first component I've already forgotten subtract away the second component so minus a2 plus b2 the second one said just add the first component a1 plus b1 2 the second component a2 plus b2 and then the third condition said take whatever the first component was and multiply it by 2 so we'll have 2 times a sub 1 plus B sub 1 and of course you could simplify this down if you want to you know so if we simplify it down a little bit we'll have a 1 plus B 1 minus a 2 minus B 2 and then well it looks like we'll just have I guess a 1 we could do we could reorder them plus a 2 plus B 1 plus B 2 doesn't really matter and then we'll have 2 a 1 plus 2 B 1 okay so this is what we got when we did when we summed the vectors first and then apply the transformation so we'll come back to that in just a second so now so we just figured out an expression for the left side let's figure out an expression for the right side as well okay so we'll do so now we'll figure out an expression T of U plus T of V thrilling stuff but again this is important linear transformations get used all the time definitely useful stuff in linear algebra then also good just you know good way if you're kind of learning proofs again just how to apply definitions so we'll apply the transformation to vector U so a1 a2 we'll do that one and then we'll add to that whatever we get after we apply the transformation to vector V okay so now I'm again I'm going to use my little rule that we started with okay so it says subtract the second component from the first it says add those components together and then it says take the first component and multiply it by two so if we apply the transformation to our first vector we'll just have a 1 minus a 2 a 1 plus a 2 to a 1 we'll do the same thing will apply the transformation to our second vector so we'll get B 1 minus B 2 we'll get B 1 plus B 2 and then it says take the first component and again double it okay so again we know how to add or subtract vectors we just do it a component at a time so a 1 minus a 2 plus B 1 minus B 2 we'll have a 1 plus a 2 plus B 1 plus B 2 and then we'll have 2 a 1 plus 2 B 1 and again this is a the right-hand side of that expression so let's see let's compare it to what we had just a second ago and see if those are in fact equal so this is what we did just a second ago let's go ahead and write it here since I kind of got cut off so the question is are these equal that's what we're trying to decide well looks like we have an a1 minus a2 plus b1 minus b2 that looks good a 1 plus a 2 plus B 1 plus B 2 that looks good and hey we have 2 a 1 plus 2 B 1 so that looks good so this first condition is in fact satisfied so we'll come back and just a second and we'll see if the other condition is also satisfied and if so we'll say hey this is in fact a linear transformation and then in a separate couple videos I'll do the exact same thing with the example in Part B

Channel: patrickJMT

Views: 649,692

Rating: 4.8688903 out of 5

Keywords: linear, transformation, algebra, vector, space, kernel, null, tutorial

Id: a9LcIKyuHQo

Length: 9min 6sec (546 seconds)

Published: Mon Sep 12 2011