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And s clearly exists because I constructed it, and we know it's well-defined because every y, for every y here, there is a solution to this. So given that I was able to find a function that these two things are true, this is by definition what it means to be invertible. Remember, so this means that f is invertible. Rem... | Proof Invertibility implies a unique solution to f(x)=y Linear Algebra Khan Academy.mp3 |
Remember, f being invertible, in order for f to be invertible, that means that there must exist some function from, so if f is a mapping from x to y, invertibility means that there must be some function, f inverse, that is a mapping from y to x such that, so I could write there exists a function, such that the inverse ... | Proof Invertibility implies a unique solution to f(x)=y Linear Algebra Khan Academy.mp3 |
We can say that s is equal to f inverse. So f is definitely invertible. So hopefully you found this satisfying. This proof is very subtle and very nuanced because we kind of keep bouncing between our sets x and y, but what we've shown is that if f is, in the beginning part of this video, we showed that if f is invertib... | Proof Invertibility implies a unique solution to f(x)=y Linear Algebra Khan Academy.mp3 |
This proof is very subtle and very nuanced because we kind of keep bouncing between our sets x and y, but what we've shown is that if f is, in the beginning part of this video, we showed that if f is invertible, then for any y there is a unique solution to the equation f of x equals y. And in the second part of the vid... | Proof Invertibility implies a unique solution to f(x)=y Linear Algebra Khan Academy.mp3 |
So the fact that both of these assumptions imply each other we can write our final conclusion of the video. That f being invertible, if f, which is a mapping from x to y is invertible, this is true if and only if, and we could write that either as a two-way arrow or we could write if for if and only if. So both of thes... | Proof Invertibility implies a unique solution to f(x)=y Linear Algebra Khan Academy.mp3 |
So I have this matrix here, this matrix A, and I guess a good place to start is let's just figure out its column space and its null space. And the column space is actually super easy to figure out. It's just the span of the column vectors of A. So we can right from the get-go write that the column space of our matrix A... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So we can right from the get-go write that the column space of our matrix A is equal to the span of the vectors 1, 2, 3, 1, 1, 1, 4, and 1, 1, 4, 1, 4, 1, and 1, 3, 2. I'm done. That was pretty straightforward. A lot easier than finding null spaces. Now, this may or may not be satisfying to you, and there's a lot of op... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
A lot easier than finding null spaces. Now, this may or may not be satisfying to you, and there's a lot of open questions. Is this a basis for the space? For example, is this a linear independent set of vectors? How can we visualize the space? And I haven't answered any of those yet. But if someone just says, hey, what... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
For example, is this a linear independent set of vectors? How can we visualize the space? And I haven't answered any of those yet. But if someone just says, hey, what's the column space of A? This is the column space of A. And now we can answer some of those other questions. If this is a linearly independent set of vec... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
But if someone just says, hey, what's the column space of A? This is the column space of A. And now we can answer some of those other questions. If this is a linearly independent set of vectors, then these vectors would be a basis for the column space of A. We don't know that yet. We don't know whether these are linear... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
If this is a linearly independent set of vectors, then these vectors would be a basis for the column space of A. We don't know that yet. We don't know whether these are linearly independent. But we can figure out if they're linearly independent by looking at the null space of A. Remember, these are linearly independent... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
But we can figure out if they're linearly independent by looking at the null space of A. Remember, these are linearly independent if the null space of A only contains the zero vector. So let's figure out what the null space of A is. And remember, we can do a little shortcut here. The null space of A is equal to the nul... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And remember, we can do a little shortcut here. The null space of A is equal to the null space of the reduced row echelon form of A. And I showed you that when we first calculated the null space of a vector. Because when you perform these, essentially if you want to solve for the null space of A, you create an augmente... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Because when you perform these, essentially if you want to solve for the null space of A, you create an augmented matrix. And you put the augmented matrix in reduced row echelon form. But the zeros never change. So essentially you're just taking A and putting it in reduced row echelon form. So let's do that. So I'll ke... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So essentially you're just taking A and putting it in reduced row echelon form. So let's do that. So I'll keep row 1 the same. 1, 1, 1, 1. And then let me replace row 2 with row 2 minus row 1. Row 2 minus row 1. So what do I get? | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
1, 1, 1, 1. And then let me replace row 2 with row 2 minus row 1. Row 2 minus row 1. So what do I get? 2. No, actually I want to zero this out here. So row 2 minus 2 times row 1. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So what do I get? 2. No, actually I want to zero this out here. So row 2 minus 2 times row 1. Actually, even better, because I eventually want to get a 1 here. So let me do 2 times row 1 minus row 2. So let me say 2 times row 1 and I'm going to minus row 2. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So row 2 minus 2 times row 1. Actually, even better, because I eventually want to get a 1 here. So let me do 2 times row 1 minus row 2. So let me say 2 times row 1 and I'm going to minus row 2. So 2 times 1 minus 2 is 0, which is exactly what I wanted there. 2 times 1 minus 1 is 1. 2 times 1 minus 4 is minus 2. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let me say 2 times row 1 and I'm going to minus row 2. So 2 times 1 minus 2 is 0, which is exactly what I wanted there. 2 times 1 minus 1 is 1. 2 times 1 minus 4 is minus 2. 2 times 1 minus 3 is minus 1. Alright. Now let me see if I can zero out this guy here. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
2 times 1 minus 4 is minus 2. 2 times 1 minus 3 is minus 1. Alright. Now let me see if I can zero out this guy here. So what can I do? Let me take, and I can do any combination, anything that essentially zeros this guy out. But I want to minimize my number of negative numbers. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now let me see if I can zero out this guy here. So what can I do? Let me take, and I can do any combination, anything that essentially zeros this guy out. But I want to minimize my number of negative numbers. So let me take this third row minus 3 times this first row. So I'm going to take minus 3 times that first row a... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
But I want to minimize my number of negative numbers. So let me take this third row minus 3 times this first row. So I'm going to take minus 3 times that first row and add it to this third row. So 3 minus 3 times 1 is 0. 4 minus 3 times 1. These are just going to be a bunch of 3's. 4 minus 3 times 1 is 1. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So 3 minus 3 times 1 is 0. 4 minus 3 times 1. These are just going to be a bunch of 3's. 4 minus 3 times 1 is 1. 1 minus 3 times 1 is minus 2. And 2 minus 3 times 1 is minus 1. Now if we want to get this into reduced row echelon form, we need to target that one there and that one there. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
4 minus 3 times 1 is 1. 1 minus 3 times 1 is minus 2. And 2 minus 3 times 1 is minus 1. Now if we want to get this into reduced row echelon form, we need to target that one there and that one there. And what can we do? So let's keep my middle row the same. My middle row is not going to change. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now if we want to get this into reduced row echelon form, we need to target that one there and that one there. And what can we do? So let's keep my middle row the same. My middle row is not going to change. 0, 1, minus 2, minus 1. And to get rid of this one up here, I can just replace my first row with my first row min... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
My middle row is not going to change. 0, 1, minus 2, minus 1. And to get rid of this one up here, I can just replace my first row with my first row minus my second row. Because then this won't change. I have 1 minus 0 is 1. 1 minus 1 is 0. That's what we wanted. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Because then this won't change. I have 1 minus 0 is 1. 1 minus 1 is 0. That's what we wanted. 1 minus minus 2 is 3. Right? That's 1 plus 2. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
That's what we wanted. 1 minus minus 2 is 3. Right? That's 1 plus 2. 1 minus minus 1, that's 1 plus 1. That is 2. Fair enough. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
That's 1 plus 2. 1 minus minus 1, that's 1 plus 1. That is 2. Fair enough. Now let me do my third row. The third row, let me just add, or let me just subtract, let me replace my third row with my third row subtracted from my first row. They're obviously the same thing. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Fair enough. Now let me do my third row. The third row, let me just add, or let me just subtract, let me replace my third row with my third row subtracted from my first row. They're obviously the same thing. So if I subtract this row from, if I subtract the third row from the second row, I'm just going to get a bunch o... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
They're obviously the same thing. So if I subtract this row from, if I subtract the third row from the second row, I'm just going to get a bunch of 0's. 0 minus 0 is 0. 1 minus 1 is 0. Minus 2 minus minus 2 is 0. And minus 1 minus minus 1, that's minus 1 plus 1. That's equal to 0. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
1 minus 1 is 0. Minus 2 minus minus 2 is 0. And minus 1 minus minus 1, that's minus 1 plus 1. That's equal to 0. And just like that, we have it now in reduced row echelon form. So this right here is the reduced row echelon form of A. That straightforward. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
That's equal to 0. And just like that, we have it now in reduced row echelon form. So this right here is the reduced row echelon form of A. That straightforward. Now, the whole reason why we even went through this exercise is we wanted to figure out the null space of A, and we already know that the null space of A is e... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
That straightforward. Now, the whole reason why we even went through this exercise is we wanted to figure out the null space of A, and we already know that the null space of A is equal to the null space of the reduced row echelon form of A. So if this is the reduced row echelon form of A, let's figure out its null spac... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So the null space is the set of all vectors in R4, because we have 4 columns here. 1, 2, 3, 4. The null space is the set of all vectors that satisfy this equation. We're going to have 3 0's right here. That's the 0 vector in R3, because we have 3 rows right there. And you can figure it out. This times this has to equal... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
We're going to have 3 0's right here. That's the 0 vector in R3, because we have 3 rows right there. And you can figure it out. This times this has to equal that 0. That dotted with that, essentially, is going to equal that 0. That dotted with that is equal to that 0. I say essentially because I didn't define a row vec... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
This times this has to equal that 0. That dotted with that, essentially, is going to equal that 0. That dotted with that is equal to that 0. I say essentially because I didn't define a row vector dot a column vector. I've only defined column vectors dotted with other column vectors, but we've covered that in a previous... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
I say essentially because I didn't define a row vector dot a column vector. I've only defined column vectors dotted with other column vectors, but we've covered that in a previous video, where we just say this is a transpose of a column vector. So let's just take this and write a system of equations with this. So we ge... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So we get 1 times x1. So this times this is going to be equal to that 0. 1 times x1, that is x1, plus 0 times x2, plus 3 times x3, plus 2 times x4 is equal to that 0. Then in yellow right here, I have 0 times x1 plus 1 times x2 minus 2 times x3 minus x4 is equal to 0. And then this gives me no information. 0's times al... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Then in yellow right here, I have 0 times x1 plus 1 times x2 minus 2 times x3 minus x4 is equal to 0. And then this gives me no information. 0's times all of this is equal to 0. So it just turns into 0 equals 0. So let's see if we can solve for our pivot entries or our pivot variables. What are our pivot entries? The l... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So it just turns into 0 equals 0. So let's see if we can solve for our pivot entries or our pivot variables. What are our pivot entries? The left of the pivot entry, that's a pivot entry. That's what reduced row echelon form is all about. Getting these entries that are 1 and they're the only non-zero term in their resp... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
The left of the pivot entry, that's a pivot entry. That's what reduced row echelon form is all about. Getting these entries that are 1 and they're the only non-zero term in their respective columns, and that every pivot entry is to the right of a pivot entry above it. And then the columns that don't have pivot entries,... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then the columns that don't have pivot entries, these columns represent the free variables. So this column has no pivot entry, and so when you take the dot product, you get this column turned into this column in our system of equations. So we know that x3 is a free variable. We can set it equal to anything. Likewis... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
We can set it equal to anything. Likewise, x4 is a free variable. x1 and x2 are pivot variables because their corresponding columns in our reduced row echelon form have pivot entries in them. So let's see if we can simplify this into a form we know, and we've seen this before. So if I solve for x1, this 0 I can ignore,... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let's see if we can simplify this into a form we know, and we've seen this before. So if I solve for x1, this 0 I can ignore, that 0 I can ignore. I could say that x1 is equal to minus 3 x3 minus 2 x4. I just subtracted these two from both sides of the equation, and I can say that x2 is equal to 2 x3 plus x4. And if... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
I just subtracted these two from both sides of the equation, and I can say that x2 is equal to 2 x3 plus x4. And if we want to write our solution set now, so if I wanted to find the null space of A, which is the same thing as the null space of the reduced row echelon form of A, is equal to all of the vectors x1, x2, x3... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And these are pivot variables because I can't just set them to anything. When I determine what my x3's and my x4's are, they determine what my x1's and my x2's have to be. So these are pivot variables, these are free variables. I can make this guy pi, and I can make this guy minus 2. We can set them to anything. So x1 ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
I can make this guy pi, and I can make this guy minus 2. We can set them to anything. So x1 is equal to, let's see, let me write it this way. They're equal to x3, let me do it in a different color, do x3 like this. So it's equal to x3 times some vector plus x4 times some other vector. So any solution set in my null spa... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
They're equal to x3, let me do it in a different color, do x3 like this. So it's equal to x3 times some vector plus x4 times some other vector. So any solution set in my null space is going to be a linear combination of these two vectors. And we can figure out what these two vectors are just from these two constraints ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And we can figure out what these two vectors are just from these two constraints right here. So let me do it in a neutral color. x1 is equal to minus 3 times x3, so minus 3 times x3, minus 2 times x4. Straightforward enough. x2 is equal to 2 times x3 plus x4. What's x3 equal to? Well x3 is equal to itself. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Straightforward enough. x2 is equal to 2 times x3 plus x4. What's x3 equal to? Well x3 is equal to itself. Whatever we said x3 equal to, that's going to be x3. So x3 is going to be 1 times x3 plus 0 times x4. It's not going to have any x4 in it. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well x3 is equal to itself. Whatever we said x3 equal to, that's going to be x3. So x3 is going to be 1 times x3 plus 0 times x4. It's not going to have any x4 in it. x3 is going to be kind of an independent variable where it's going to be free. We can set it to whatever it is. So we set it and then that's going to be ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
It's not going to have any x4 in it. x3 is going to be kind of an independent variable where it's going to be free. We can set it to whatever it is. So we set it and then that's going to be our x3 in our solution set. x4 is not going to have any x3 in it. It's just going to be 1 times x4. And so our null space is essen... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So we set it and then that's going to be our x3 in our solution set. x4 is not going to have any x3 in it. It's just going to be 1 times x4. And so our null space is essentially all of the linear combinations of these two vectors. This can be any real number. This is just any real number and x4 is just any member of th... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And so our null space is essentially all of the linear combinations of these two vectors. This can be any real number. This is just any real number and x4 is just any member of the real space. So all of these, the set of all of the valid solutions to Ax is equal to 0. Where did I write that? Did I even write that down?... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So all of these, the set of all of the valid solutions to Ax is equal to 0. Where did I write that? Did I even write that down? No, I haven't even written that anywhere. The set of all Ax is equal to 0 where this is my x, it equals all the linear combinations of this vector and that vector right there. And we know what... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
No, I haven't even written that anywhere. The set of all Ax is equal to 0 where this is my x, it equals all the linear combinations of this vector and that vector right there. And we know what all of the linear combinations mean. It means my null space is equal to the span of these two guys. The span of minus 3, 2, 1, ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
It means my null space is equal to the span of these two guys. The span of minus 3, 2, 1, 0 and minus 2, 1, 0, 1. Now, let me ask you a question. Are the columns in A, are they a linearly independent set? Let me write that down. So if we write these vectors right there, these are the vectors, the column vectors of A. S... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Are the columns in A, are they a linearly independent set? Let me write that down. So if we write these vectors right there, these are the vectors, the column vectors of A. So let me write that down. So are the column vectors of A, what were they? 1, 2, 3, 1, 1, 4, 1, 4, 1 and 1, 3, 2. So this is just the column vector... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let me write that down. So are the column vectors of A, what were they? 1, 2, 3, 1, 1, 4, 1, 4, 1 and 1, 3, 2. So this is just the column vectors of A. I could just write A as just this bunch of columns. But my question is, is this a linearly independent set? And here you might immediately start thinking, well, when... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So this is just the column vectors of A. I could just write A as just this bunch of columns. But my question is, is this a linearly independent set? And here you might immediately start thinking, well, when we said that something is linearly independent, so linearly independence, let me just write it like this. Linear ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Linear independence implies that there's only one solution. We saw this, I think, two videos ago that there's only one solution. One solution to Ax is equal to 0, and that is the 0 solution, that x is equal to the 0 vector. Or another way to say that is that the null space of my matrix A is equal to just the 0 vector. ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Or another way to say that is that the null space of my matrix A is equal to just the 0 vector. That's what linear independence implies, and it goes both ways. If my null space is just the 0 vector, then I know it's linearly independent. If my null space includes other vectors, then I am not linearly independent. Now, ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
If my null space includes other vectors, then I am not linearly independent. Now, my null space of A, what does it include? Is it just the 0 vector? Well, no, it includes every linear combination of these guys. It includes actually an infinite number of vectors. It's not just one solution. Obviously, 0 vector is contai... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well, no, it includes every linear combination of these guys. It includes actually an infinite number of vectors. It's not just one solution. Obviously, 0 vector is contained here. If you just multiply both of these, if you pick 0 for that and that, it's contained. But you can get a whole set of vectors. Because the sp... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Obviously, 0 vector is contained here. If you just multiply both of these, if you pick 0 for that and that, it's contained. But you can get a whole set of vectors. Because the span of this guy, so you can kind of, you know, the null span of A, the null space, sorry, the null space of A does not just contain the 0 vecto... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Because the span of this guy, so you can kind of, you know, the null span of A, the null space, sorry, the null space of A does not just contain the 0 vector. So it has more than just 0. So what does that mean? Well, that means that there's more than one solution to this, and that means that this is a linearly dependen... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well, that means that there's more than one solution to this, and that means that this is a linearly dependent set. And what does that mean? Well, at the very beginning of the video, I said, what's the column space of A? And we said, oh, the column space of A is just the span of the column vectors, right? I just wrote ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And we said, oh, the column space of A is just the span of the column vectors, right? I just wrote it out like that. And I said, well, it's not clear whether this is a valid basis for the column space of A. And what's a basis? A basis is a set of vectors that span a subspace, and they are also linearly independent. And... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And what's a basis? A basis is a set of vectors that span a subspace, and they are also linearly independent. And we just showed that these guys are not linearly independent. So that means that they are not a basis for the column space of A. They do span the column space of A, by definition, really, but they're not a b... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So that means that they are not a basis for the column space of A. They do span the column space of A, by definition, really, but they're not a basis. They're just linearly independent for them to be a basis. So let's see if we can figure out what a basis for this column space would be. And to do that, we just have to ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let's see if we can figure out what a basis for this column space would be. And to do that, we just have to get rid of some redundant vectors. If I can show you that this guy can be represented by some combination of these two guys, then I can get rid of that guy. He's not adding any new information. Same with that ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
He's not adding any new information. Same with that guy. Who knows? So let's see if we can figure this piece of the puzzle out. We know already that x1 times 1, 2, 3, plus x2 times 1, 1, 4, plus x3 times 1, 4, 1, plus x4 times 1, 3, 2. We know that this is equal to 0. Now, if we're able to solve for x4 in terms of, let... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So let's see if we can figure this piece of the puzzle out. We know already that x1 times 1, 2, 3, plus x2 times 1, 1, 4, plus x3 times 1, 4, 1, plus x4 times 1, 3, 2. We know that this is equal to 0. Now, if we're able to solve for x4 in terms of, let's say I can solve them in terms of, let me just think that I can so... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now, if we're able to solve for x4 in terms of, let's say I can solve them in terms of, let me just think that I can solve for the vectors that are associated with my free variables using the other vectors. Let me see if I can do that, and you'll see it's actually pretty straightforward. So let's say I want to figure o... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So if I subtract this from both sides of this equation, I get what? I get minus, let me put it this way, let me set x3 equal to 0. It was a free variable. I can do that. So if I set x3 is equal to 0, then what do I get here? I get, if I set x3 equal to 0, this guy disappears. And if I subtract this from both sides of t... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
I can do that. So if I set x3 is equal to 0, then what do I get here? I get, if I set x3 equal to 0, this guy disappears. And if I subtract this from both sides of this equation, I get x1 times 1, 2, 3, plus x2 times 1, 1, 4 is equal to, I'm just setting x3 equal to 0. That was a free variable. So I'm setting x3 equal ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And if I subtract this from both sides of this equation, I get x1 times 1, 2, 3, plus x2 times 1, 1, 4 is equal to, I'm just setting x3 equal to 0. That was a free variable. So I'm setting x3 equal to 0, so this whole thing disappears. So that is equal to minus x4 times 1, 3, 2. Now I set x3 equal to 0, let me set x1, ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So that is equal to minus x4 times 1, 3, 2. Now I set x3 equal to 0, let me set x1, sorry, let me set x4 to be equal to minus 1. x4 is equal to minus 1. If x4 is equal to minus 1, what is minus x4? Well then this thing will just be equal to 1. And I'll have x1 times 1, 2, 3 plus x2 times 1, 1, 4 will equal this fourth ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well then this thing will just be equal to 1. And I'll have x1 times 1, 2, 3 plus x2 times 1, 1, 4 will equal this fourth vector right here. And can I always find things like this? Well sure, I can actually find the particular ones. If x3 is equal to 0 and x4 is minus 1, let me copy and paste this that I have up here. ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well sure, I can actually find the particular ones. If x3 is equal to 0 and x4 is minus 1, let me copy and paste this that I have up here. Let me copy and paste this. Sometimes it doesn't. Edit, copy, edit, paste. There you go. Let me scroll down a little bit. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Sometimes it doesn't. Edit, copy, edit, paste. There you go. Let me scroll down a little bit. This is what we got when we figured out our null space right there. So, if I'm setting, remember these are the free variables. If I set x3 equal to 0 and x4 is equal to minus 1, what is x1? | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Let me scroll down a little bit. This is what we got when we figured out our null space right there. So, if I'm setting, remember these are the free variables. If I set x3 equal to 0 and x4 is equal to minus 1, what is x1? Then this will imply that x1 is equal to minus 3 times x3, that's just 0, minus 2 times x4. If x4... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
If I set x3 equal to 0 and x4 is equal to minus 1, what is x1? Then this will imply that x1 is equal to minus 3 times x3, that's just 0, minus 2 times x4. If x4 is minus 1, minus 2 times minus 1, x1 will equal 2. And then what will x2 be equal to? x2 is equal to 2 times x3, which is 0, plus x4. So it's equal to minus 1... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And then what will x2 be equal to? x2 is equal to 2 times x3, which is 0, plus x4. So it's equal to minus 1. I just showed you that if I set this equal to 2 and this equal to minus 1, I have a linear combination of this vector and this vector that can add up to this fourth vector. You can even verify it. 2 times 1 minu... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
I just showed you that if I set this equal to 2 and this equal to minus 1, I have a linear combination of this vector and this vector that can add up to this fourth vector. You can even verify it. 2 times 1 minus 1 is equal to 1. 2 times 2 minus 1 is equal to 3. 2 times 3 is 6. Minus 4 is equal to 2. So it checks out. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
2 times 2 minus 1 is equal to 3. 2 times 3 is 6. Minus 4 is equal to 2. So it checks out. I just showed you using really our definitions, really looking at what were our free variables versus our pivot variables, we were able to show you, kind of just very simply solve for this fourth vector in terms of these first two... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So it checks out. I just showed you using really our definitions, really looking at what were our free variables versus our pivot variables, we were able to show you, kind of just very simply solve for this fourth vector in terms of these first two. We know, if we go back to the set, that this fourth vector is really u... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now let's see if this guy, this third guy, we can do the same exercise. This is also dictated by a free variable. So let's see if I can write him as a combination of these first two. Well, we'll do the exact same thing. Instead of setting x3 equal to 0 and x4 equal to minus 1, let's set, let us set x4 is equal to 0 bec... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Well, we'll do the exact same thing. Instead of setting x3 equal to 0 and x4 equal to minus 1, let's set, let us set x4 is equal to 0 because I want to cross that out. And let me set x3 is equal to minus 1. If x3 is equal to minus 1, so this equals minus 1, what is our, what is this equation reduced to? We get x1 times... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
If x3 is equal to minus 1, so this equals minus 1, what is our, what is this equation reduced to? We get x1 times 1, 2, 3 plus x2 times 1, 1, 4 is equal to, if this is minus 1 times 1, 4, 1 and then we add it to both sides of this equation, we get plus 1 times 1, 4, 1. And once again, we can just solve for our x1's and... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
If x4 is 0 and x3 is minus 1, then x1, x4 is 0, so x3 is just minus 3 times x3. So x1 would be equal to 3, right? Minus 3 times minus 1. And what would x2 be equal to? x4 is 0. We can ignore that. x2 would be equal to minus 2. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
And what would x2 be equal to? x4 is 0. We can ignore that. x2 would be equal to minus 2. So this would be 3 and then this would be minus 2. Let's see if it works out. 3 times 1 minus 2 is 1. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
x2 would be equal to minus 2. So this would be 3 and then this would be minus 2. Let's see if it works out. 3 times 1 minus 2 is 1. 3 times 2 minus 2 is 4. 3 times 3 minus 8 is 1. It checks out. | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
3 times 1 minus 2 is 1. 3 times 2 minus 2 is 4. 3 times 3 minus 8 is 1. It checks out. So I'm able to write this vector that was associated with a free variable as a linear combination of these two. So we can get rid of him from our set. So now I've shown that this guy can be written as a linear combination of these tw... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
It checks out. So I'm able to write this vector that was associated with a free variable as a linear combination of these two. So we can get rid of him from our set. So now I've shown that this guy can be written as a linear combination of these two. So the span of all of those guys should be equal to the span. So let ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
So now I've shown that this guy can be written as a linear combination of these two. So the span of all of those guys should be equal to the span. So let me write it this way. The column space of A, I can now rewrite. Before it was the span of all of those vectors. It was the span of all of the column vectors, v1, v2, ... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
The column space of A, I can now rewrite. Before it was the span of all of those vectors. It was the span of all of the column vectors, v1, v2, v3, and v4. Now I just showed you that v3 and v4 can be rewritten in terms of v1 and v2. So they're redundant. So that is equal to the span of v1 and v2, which are just those t... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
Now I just showed you that v3 and v4 can be rewritten in terms of v1 and v2. So they're redundant. So that is equal to the span of v1 and v2, which are just those two vectors. One vector 1, 2, 3 and vector 1, 1, 4. Now, are any of these guys redundant? Can I express one of them as a linear combination of the other? And... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
One vector 1, 2, 3 and vector 1, 1, 4. Now, are any of these guys redundant? Can I express one of them as a linear combination of the other? And essentially when I'm talking about the linear combination of only one other vector, it's just multiplying it by a scalar. Well, let's think about that. There's multiple ways y... | Null space and column space basis Vectors and spaces Linear Algebra Khan Academy.mp3 |
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