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So let me write this x as a yellow x. And let me write this whole term as x plus y. So this right here can be rewritten as x dot, so this x dot this x plus y. And then it would be plus this y dot, I wanted to switch colors, plus this y dot x plus y. It's good to see that when you're multiplying these, you're just apply... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then it would be plus this y dot, I wanted to switch colors, plus this y dot x plus y. It's good to see that when you're multiplying these, you're just applying the distributive property. All I did is I distributed this term along each of these terms in this sum right here. So then I got this. And then I can distri... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
So then I got this. And then I can distribute each of these into the sum. So then this becomes, I'll be careful with the colors, x dot x plus x dot y. Maybe this was a little bit overkill, but I think it's good to see that this isn't just some magic here. And we're just using the exact properties that we proved of the ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Maybe this was a little bit overkill, but I think it's good to see that this isn't just some magic here. And we're just using the exact properties that we proved of the dot product. So that's that right there. And then it's plus y dot x. So y dot x plus this yellow y dot the blue y. So the magnitude or the length of ou... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then it's plus y dot x. So y dot x plus this yellow y dot the blue y. So the magnitude or the length of our vector x plus y squared can be rewritten like this. And this, and I'll just switch back to one color. So this equals that. And all of that, what does this equal? This is equal to x dot x. | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And this, and I'll just switch back to one color. So this equals that. And all of that, what does this equal? This is equal to x dot x. So this is equal to, what's x dot x? x dot x is just the magnitude. So let me write this is just equal to the magnitude of our vector x. I should stop using the word magnitude. | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
This is equal to x dot x. So this is equal to, what's x dot x? x dot x is just the magnitude. So let me write this is just equal to the magnitude of our vector x. I should stop using the word magnitude. The length of our vector x squared. And then I have two terms here. I have an x dot y and a y dot x. | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let me write this is just equal to the magnitude of our vector x. I should stop using the word magnitude. The length of our vector x squared. And then I have two terms here. I have an x dot y and a y dot x. But we know that x dot y and y dot x are really the same thing. We showed that order doesn't matter when you t... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
I have an x dot y and a y dot x. But we know that x dot y and y dot x are really the same thing. We showed that order doesn't matter when you take the dot product, just like it doesn't matter with regular multiplication. So these are really the same terms. So we could write plus 2 times x dot y. And then finally we hav... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
So these are really the same terms. So we could write plus 2 times x dot y. And then finally we have that last term sitting here. We have this y dot y. y dot y is the same thing as the length of our vector y squared. Now, let's see if we can break out our Cauchy-Schwarz inequality. Or maybe Schwarz, I don't know if I'm... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
We have this y dot y. y dot y is the same thing as the length of our vector y squared. Now, let's see if we can break out our Cauchy-Schwarz inequality. Or maybe Schwarz, I don't know if I'm pronouncing it right. But x dot y. So x dot y, we have the absolute value of x dot y here. But we know that just x dot y is going... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
But x dot y. So x dot y, we have the absolute value of x dot y here. But we know that just x dot y is going to be, it has to be less than or equal to the absolute value of x dot y. Why is that? Well, this could be negative. I could show you examples of dot products that are negative. In fact, if x has all positive term... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Why is that? Well, this could be negative. I could show you examples of dot products that are negative. In fact, if x has all positive terms and y has all negative terms, you're going to have a negative dot product. So this could be negative. Or it could be positive. If it's positive, the absolute value, they're equal ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
In fact, if x has all positive terms and y has all negative terms, you're going to have a negative dot product. So this could be negative. Or it could be positive. If it's positive, the absolute value, they're equal to each other. If this is negative, then this absolute value is definitely going to be greater than it. ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
If it's positive, the absolute value, they're equal to each other. If this is negative, then this absolute value is definitely going to be greater than it. So we can add to the Cauchy-Schwarz inequality. And this is a bit obvious. We could say, look, we could add a little x dot y is less than or equal to the absolute v... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And this is a bit obvious. We could say, look, we could add a little x dot y is less than or equal to the absolute value of x dot y, which is less than or equal to the length of x times the length of y. So x dot y, the dot product of x with y is definitely less than its absolute value of that, which is definitely less ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
So if I rewrite this, this statement right here is definitely less than or equal to this exact statement. But if I replace these with the lengths of the vectors, so that is definitely less than or equal to, I'm just rewriting this x squared, and I'll write the plus 2 there. But I want to make it very clear what I'm rep... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then I have the plus length of my vector y there squared. Now this, I'm saying, this is definitely less than the absolute value of x dot y, which is definitely less by the Cauchy-Schwarz inequality, definitely less than the product of the two lengths. So all I did here, so I'm just replacing this with the product o... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
So I'm going to put x, the length of x times the length of y, right? And since this is the same as that, this is the same as that, but this is definitely less than this. This whole term has to be less than this whole term. Now let me just remind you what we were doing. I said that this thing that I wrote over here, thi... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now let me just remind you what we were doing. I said that this thing that I wrote over here, this is the same as that. So this thing up here, which is the same as that, which is less than that, also. So we can write that the magnitude of x plus y squared, and not the magnitude, the length of the vector x plus y square... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
So we can write that the magnitude of x plus y squared, and not the magnitude, the length of the vector x plus y squared is less than this whole thing that I wrote out here. Now, or less than or equal to. Now what is this thing? Remember, I mean this might look all fancy with my little double lines around everything, b... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Remember, I mean this might look all fancy with my little double lines around everything, but these are just numbers. This length of x squared, this is just a number. Each of these are numbers. And I can just say, hey, look, this looks like a perfect square to me. This looks just like this term on the right-hand side i... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And I can just say, hey, look, this looks like a perfect square to me. This looks just like this term on the right-hand side is the exact same thing as the length of x plus the length of y plus the length of y, everything squared, right? If you just square this out, you'll get x squared plus 2 times the length of x tim... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And if we just take the square root of both sides of this, you get the length of our vector x plus y is less than or equal to the length of the vector x by itself plus the length of the vector y. And we call this the triangle inequality, which you might have remembered from geometry. Triangle inequality. Now, why is it... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now, why is it called a triangle inequality? Well, you can imagine each of these to be separate sides of a triangle. In fact, let's draw it. We can draw this in R2. Let me turn my graph paper on. Let me see where the graphs show up. If I turn my graph paper on right there, maybe I'll draw it here. | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
We can draw this in R2. Let me turn my graph paper on. Let me see where the graphs show up. If I turn my graph paper on right there, maybe I'll draw it here. So let's draw my vector x. So let's say that my vector x looks something like this. Let's say that's my vector x. | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
If I turn my graph paper on right there, maybe I'll draw it here. So let's draw my vector x. So let's say that my vector x looks something like this. Let's say that's my vector x. It's the vector 2, 4. So that's my vector x. And let's say my vector y, well, I'm just going to do it head to tail because I'm going to add ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let's say that's my vector x. It's the vector 2, 4. So that's my vector x. And let's say my vector y, well, I'm just going to do it head to tail because I'm going to add the two. So my vector y, I'm going to do it in non-standard position. Let's say my vector y looks something like this. I'm going to draw it properly. | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And let's say my vector y, well, I'm just going to do it head to tail because I'm going to add the two. So my vector y, I'm going to do it in non-standard position. Let's say my vector y looks something like this. I'm going to draw it properly. So let's say that's my vector y. Now, what does x plus y look like? x plus ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
I'm going to draw it properly. So let's say that's my vector y. Now, what does x plus y look like? x plus y, and remember, I can't necessarily draw any two vectors on two-dimensional space like this. I'm just assuming that these are in R2. But this is just to give you the idea. So then this is their sum, right? | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
x plus y, and remember, I can't necessarily draw any two vectors on two-dimensional space like this. I'm just assuming that these are in R2. But this is just to give you the idea. So then this is their sum, right? From the tail of x to the head of y. So this vector right here is the vector x plus y. And that's why it's... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
So then this is their sum, right? From the tail of x to the head of y. So this vector right here is the vector x plus y. And that's why it's called a triangle inequality. It's just saying that, look, this thing is always going to be less than or equal to, or the length of this thing is always going to be less than or e... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And that's why it's called a triangle inequality. It's just saying that, look, this thing is always going to be less than or equal to, or the length of this thing is always going to be less than or equal to the length of this thing plus the length of this thing. And that's kind of obvious when you just learn two-dimens... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
This is a much more efficient way of getting from this point to this point than going out here and then going out here. And then what's the limiting, or what's the case in which this length is equal to these lengths? Well, if you keep flattening this triangle out and you go to the extreme case where maybe the vector x ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now x plus y will just be this whole vector. Now that whole thing is x plus y. And this is the case now where the triangle inequality turns into an equality. That's why that little equal sign there. The extreme case where essentially x and y are collinear. And why does that work out? We can even go back to our math and... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
That's why that little equal sign there. The extreme case where essentially x and y are collinear. And why does that work out? We can even go back to our math and understand that. Let me turn my graph off. We can go back to our math here. If I go back to this point, remember right here I made the statement, look, this ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
We can even go back to our math and understand that. Let me turn my graph off. We can go back to our math here. If I go back to this point, remember right here I made the statement, look, this thing is definitely less than this thing over here. But what if I made an assumption? What if I said that x is equal to some sc... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
If I go back to this point, remember right here I made the statement, look, this thing is definitely less than this thing over here. But what if I made an assumption? What if I said that x is equal to some scalar times y? And actually, I have to be careful, because just some scalar times y, remember our Cauchy-Schwarz ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And actually, I have to be careful, because just some scalar times y, remember our Cauchy-Schwarz inequality said that look, the inequality turns into an equality if x is some non-zero scalar of y. And then we can apply this. We can say that the absolute value of x dot y is the same as this over here. But I don't have ... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
But I don't have the absolute value of x dot y here. I don't know that this is positive. I can say definitively that this is a positive quantity because I took the absolute value of it. No absolute value here. So the only way that I can ensure that this is a positive quantity, that this is the same thing as the absolut... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
No absolute value here. So the only way that I can ensure that this is a positive quantity, that this is the same thing as the absolute value of x dot y, is to enforce, if I'm going to go down this road, is to enforce that this term right here, that c be positive. Because if c is positive, then x dot y, then that would... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And the only way that I can ensure that this expression right here is equal to the absolute value of x dot y, the only way I can assure this is if c is positive. If c is negative, then this is going to be a negative number while this is a positive. So if I assume that this is positive, then I can say that x dot y is eq... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And since it's a scalar multiple, then I could say that that term is equal to, not just less than or equal to, the magnitude of x's and y's. So hopefully I'm not confusing you. So all I'm saying is, if I can assume that x is some positive scalar multiple of y, that this wouldn't be a less than sign, then I could say th... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And if it's the same thing as the absolute value of x dot y, and it's some scalar multiple of each other, then we could go down this other route. We could say that this thing here, I don't want to get too messy, we could say that this is equal to that. If this is equal to that, then this would have become an equal sign... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then we would have had the limiting case. But we could say that x plus y, we've done the same work, but we'd had an equal sign the whole way, would equal the length of x plus y. In the situation where x is equal to some positive scalar times y, so c is greater than 0. These two imply each other. And we saw that geo... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
These two imply each other. And we saw that geometrically. I lost my axes here, but we see that the only time that the length of x plus y is equal to the length of x plus the length of y is when they're collinear. Over here, this plus this is clearly, you can just visually look at it, longer than this right there. So y... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Over here, this plus this is clearly, you can just visually look at it, longer than this right there. So you might be saying, Sal, once again, this linear algebra is a little bit silly. We learned the triangle inequality in eighth or ninth grade. Why did you go through all of this pain to redefine it? And this is the i... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Why did you go through all of this pain to redefine it? And this is the interesting thing. What I just drew here, and this is what you learned in ninth grade geometry, this is just an R2. This is just your Cartesian coordinates. Or I don't want to use the word dimension too much, because we're going to define that form... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
This is just your Cartesian coordinates. Or I don't want to use the word dimension too much, because we're going to define that formally, but this is kind of your two-dimensional space that's going on. But what's interesting or what's useful about linear algebra is we've just defined the triangle inequality for arbitra... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Each of these, these don't have to be an R2. These could be, this is true if we're in R100, where every vector has 100 components to it. We've just defined some notion of the triangle inequality. We've abstracted well beyond just our two-dimensional Cartesian coordinates, well beyond even our three dimensions, to essen... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
We've abstracted well beyond just our two-dimensional Cartesian coordinates, well beyond even our three dimensions, to essentially n-dimensional space. And I haven't defined dimensions yet, but I think you're starting to appreciate what they are. But anyway, hopefully you found that useful. And we can now take this res... | Vector triangle inequality Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now let's say we know that the linear transformation T can be this transformation matrix, when you put it in reduced row echelon form, the transformation matrix, it is equal to an n by n identity matrix, or the n by n identity matrix. Well, this alone tells us a lot of things. First of all, if when you put this in redu... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
And we saw in the last video, all of these are conditions, especially this one right here, for T to be invertible. So if we know that this is true, T is a linear transformation, it's the reduced row echelon form of its transformation matrix, is the identity matrix right there. We know that T is invertible. But let's re... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
But let's remind ourselves what it even means to be invertible. To be invertible means that there exists some, we used the word function before, but now we're talking about transformations, they're really the same thing. So let's say there exists some transformation, let's call it T inverse, T inverse like that, or T t... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
And just to remind you what this looks like, let's draw our domains and codomain. Our domain is Rn, and our codomain is also Rn. So it's just like that. Codomain is also Rn. So if you take some vector in your domain, apply the transformation T, you're going to go into your codomain, so that is T. And then if you apply ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Codomain is also Rn. So if you take some vector in your domain, apply the transformation T, you're going to go into your codomain, so that is T. And then if you apply the T inverse after that, you're going to go back to that original x. So this says, look, you apply T and then you apply T inverse, you're just going to ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
It's equivalent to the identity transformation, just like that. This is saying, if you start here in your codomain, you apply your inverse first, you apply this inverse transformation first, then you apply your transformation, you're going to go back to the same point in your codomain. So it's equivalent to the identit... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
It just happens to be in this case that the domain and the codomain are the same set, Rn. Now, we know what a transformation, what it means to be invertible. We know what the conditions are for invertibility. So this begs the next question. We know that this guy is a linear transformation. In fact, that's one of the co... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So this begs the next question. We know that this guy is a linear transformation. In fact, that's one of the conditions to be able to represent it as a matrix. Or any transformation that can be represented as a matrix vector product is a linear transformation. So this guy is a linear transformation. But the question is... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Or any transformation that can be represented as a matrix vector product is a linear transformation. So this guy is a linear transformation. But the question is, is T inverse a linear transformation? Now, let's review what the two conditions are that we need to have to be a linear transformation. So we know T is a line... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Now, let's review what the two conditions are that we need to have to be a linear transformation. So we know T is a linear transformation. So we know that if you apply the transformation T to two vectors, let's say x and y, if we apply it to the sum of those two vectors, it is equal to the transformation of the first v... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
That's one of the conditions, or one thing that we know is true for all linear transformations. And the second thing that we know is true for all linear transformations is if we take the transformation of some scaled version of a vector in our domain, it is equal to the scaling factor times the transformation of the ve... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So let's see if we can prove that both of these conditions hold for T inverse, for this guy right here. So to do this, let's do this little exercise right here. Let's apply T, let's take the composition of T with T inverse of two vectors, a plus b. Remember, T inverse is a mapping from your codomain to your domain, alt... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Remember, T inverse is a mapping from your codomain to your domain, although they're both going to be Rn in this case, but T inverse maps from this set to that set. Let's write it up here. T inverse is a mapping from your codomain to your domain, although it looks identical just like that. OK, so what is this going to ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
OK, so what is this going to be equal to? Well, we just said, by definition, of your inverse transformation, this is going to be equal to the identity transformation on your codomain. So assuming that these guys are members of your codomain, in this case, Rn, this is just going to be equal to a plus b. This thing, the ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
This thing, the composition of T with its inverse, by definition, is just the identity transformation on your codomain. So this is just whatever I put in here. If I put in an x here, this would be an x. If I put in an apple here, this would be an apple. It's going to be the identity transformation. Now what is this equ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
If I put in an apple here, this would be an apple. It's going to be the identity transformation. Now what is this equal to? What is this equal to? Well, I could use the same argument to say that this right here is equal to the identity transformation applied to a. And I'm not writing the identity transformation. I'm wr... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
What is this equal to? Well, I could use the same argument to say that this right here is equal to the identity transformation applied to a. And I'm not writing the identity transformation. I'm writing this, but we know that this is equivalent to the identity transformation. So we could say that equivalent to the compo... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
I'm writing this, but we know that this is equivalent to the identity transformation. So we could say that equivalent to the composition of T with the inverse applied to a. And we could say that this is equivalent to the identity transformation, which we know is the same thing as the composition of T with T inverse app... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So we can rewrite this thing right here as being equal to the sum of these two things. In fact, we don't even have to rewrite it. We can just write it equal to. This transformation is equal to this. And maybe an easier way for you to, I guess, process it is, we could write this as T of the T inverse of a plus b is equa... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
This transformation is equal to this. And maybe an easier way for you to, I guess, process it is, we could write this as T of the T inverse of a plus b is equal to T of the T inverse of a plus T of the T inverse of b. And this should, I don't know which one your brain processes easier, but either of these, when you tak... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
You take the composition of T with T inverse, you're left with an a. You take the composition of T with T inverse, you're just left with a b there. So in either case, you get a plus b, the vector a plus, when you evaluate either side of this expression, you'll get the vector a plus the vector b. Now what can we do? We ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Now what can we do? We know that T itself is a linear transformation. And since T is a linear transformation, we know that T applied to the sum of two vectors is equal to T applied to each of those vectors and summed up. Or we could take it the other way. T applied to two separate vectors, so that we could call this on... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Or we could take it the other way. T applied to two separate vectors, so that we could call this one vector right here and this vector right here. So in this case, I have a T applied to one vector, and I'm summing it to a T applied to another vector. So it's this right here, which we know is equal to T applied to the s... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So it's this right here, which we know is equal to T applied to the sum of those two vectors. So this is T applied to the vector T inverse of a. Let me write it here. So this is going to be equal to T inverse of a plus T inverse of b. It might look a little convoluted, but all I'm saying is, look, this looks just like ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So this is going to be equal to T inverse of a plus T inverse of b. It might look a little convoluted, but all I'm saying is, look, this looks just like this. If you say that x is equal to T inverse of a, and if you say that y is equal to T inverse of b. So if this looks just like that, it's going to be equal to the tr... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So if this looks just like that, it's going to be equal to the transformation T applied to the sum of those two vectors. So it's going to be equal to the transformation T applied to the inverse of a, T inverse of a, plus T inverse of b. I just use the fact that T is linear to get here. Now what can I do? Let me simplif... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Let me simplify everything that I've written right here. So I now have, let me rewrite this. This thing up here, which is the same thing as this, T, the composition of T with T inverse applied to a plus b is equal to the composition, or actually not the composition, just T applied to two vectors. T inverse of a plus T ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
T inverse of a plus T inverse of vector b. That's what we've gotten so far. Now we're very close to proving that this condition is true for T inverse. If we can just get rid of these Ts. Well the best way to get rid of those Ts is to take the composition with T inverse on both sides, or take the T inverse transformatio... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
If we can just get rid of these Ts. Well the best way to get rid of those Ts is to take the composition with T inverse on both sides, or take the T inverse transformation of both sides of this equation. So let's do that. So let's take T inverse of this side, T inverse of that side, should be equal to T inverse of this ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So let's take T inverse of this side, T inverse of that side, should be equal to T inverse of this side. Because these two things are the same thing, so if you put the same thing into a function, you should get the same value on both sides. So what is this thing on the left-hand side? What is this? This is the composit... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
What is this? This is the composition, let me write it this way, this is the composition of T inverse with T, that part, applied to this thing right here. I'm just changing the associativity of this. Applied to T inverse of the vector a plus the vector b. That's what this left-hand side is. This part right here, T inve... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Applied to T inverse of the vector a plus the vector b. That's what this left-hand side is. This part right here, T inverse of T of this, this first two steps, I'm just writing as the composition of T inverse with T, applied to this right here. That right there is the same thing as that right there. So that was another... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
That right there is the same thing as that right there. So that was another way to write that. And so that is going to be equal to the composition of T inverse with T, so I'll write it in the same color. The composition of T inverse with T, that's this part right here, which is very similar to that part right there, of... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
The composition of T inverse with T, that's this part right here, which is very similar to that part right there, of this stuff right here. Of T inverse of a plus T inverse of the vector b. Now, by definition of what T inverse is, what is this? This is the identity transformation on our domain. This is the identity tra... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
This is the identity transformation on our domain. This is the identity transformation on Rn. This is also the identity transformation on Rn. So if you apply the identity transformation to anything, you're just going to get anything. So this is going to be equal to, and I'll do it on both sides of the equation, this wh... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So if you apply the identity transformation to anything, you're just going to get anything. So this is going to be equal to, and I'll do it on both sides of the equation, this whole expression on the left-hand side is just going to simplify to the T inverse of the vectors a plus the vector b. And the right-hand side is... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
T inverse of the vector a plus T inverse of the vector b. And just like that, T inverse has met its first condition for being a linear transformation. It's met its first condition. Now let's see if we can do the second condition. Let's do the same type of thing. Let's take the composition of T with T inverse of, let's ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Now let's see if we can do the second condition. Let's do the same type of thing. Let's take the composition of T with T inverse of, let's take the composition of that, on some vector, let's call it CA, just like that. Well, we know what this is equal to. This is equal to the identity transformation on Rn. So this is j... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Well, we know what this is equal to. This is equal to the identity transformation on Rn. So this is just going to be equal to CA. Now, what is A equal to? What is A equal to? What is this thing right there equal to? I'll write it on the side right here. | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
Now, what is A equal to? What is A equal to? What is this thing right there equal to? I'll write it on the side right here. Let me do it in an appropriate color. We could say that A, the vector A is equal to the transformation T with the composition of T with T inverse, T inverse applied to the vector A, because this i... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
I'll write it on the side right here. Let me do it in an appropriate color. We could say that A, the vector A is equal to the transformation T with the composition of T with T inverse, T inverse applied to the vector A, because this is just the identity transformation. So we can rewrite this expression here as being eq... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So we can rewrite this expression here as being equal to C times the composition of T with T inverse applied to my vector A. And maybe it might be nice to rewrite it in this form instead of this composition form. So this left expression, we can just write it as saying T of the T inverse of C times the vector A. All I d... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
All I did is rewrite this left-hand side this way, is equal to this green thing right here. Well, I'll rewrite similarly. This is equal to C times T, the transformation T applied to the transformation T inverse applied to A. This is by definition what composition means. Now, T is a linear transformation, which means th... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
This is by definition what composition means. Now, T is a linear transformation, which means that if you take C times T times some vector, that is equivalent to T applied to C times that vector. This is one of the conditions of a linear transformation. So this is always going to be the case with T. So this is some vect... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So this is always going to be the case with T. So this is some vector that T is applying to. This is some scalar. So this thing, because we know that T is a linear transformation, we can rewrite as being equal to T applied to the scalar C times T inverse applied to A. And now what can we do? Well, let's apply the T inv... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
And now what can we do? Well, let's apply the T inverse transformation to both sides of this. Let me rewrite it. So on this side, we get T of T inverse of CA is equal to T of C times T inverse times A. That's what we have so far. And wouldn't it be nice if we could get rid of these outer Ts? And the best way to do that... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
So on this side, we get T of T inverse of CA is equal to T of C times T inverse times A. That's what we have so far. And wouldn't it be nice if we could get rid of these outer Ts? And the best way to do that is to take the T inverse transformation of both sides. So let's do that. T inverse, let's take that of both side... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
And the best way to do that is to take the T inverse transformation of both sides. So let's do that. T inverse, let's take that of both sides of this equation. T inverse of both sides. And another way that this could be written, this is equivalent to the composition of T inverse with T applied to C times our vector A. ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
T inverse of both sides. And another way that this could be written, this is equivalent to the composition of T inverse with T applied to C times our vector A. This right here, I've just decided to keep writing it in this form. And I took these two guys out and I wrote them as a composition. And this on the right-hand ... | Showing that inverses are linear Matrix transformations Linear Algebra Khan Academy.mp3 |
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