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Current time:0:00Total duration:10:23

Introduction to the null space of a matrix

Video transcript

let's review our notions of subspaces again and then let's see if we can define some interesting subspaces dealing with matrices and vectors so a subspace let me--just subspace let's say that I have some subspace oh let me just call it some subspace s this is a subspace if the following are true and this is all a review that the zero vector I'll just do it like that the zero vector is a member of s what contains the zero vector that if v1 and v2 are both members of my subspace then v1 plus v2 is also a member of my subspace so that's just saying that the subspaces are closed under addition you can add any of their two members and you'll get another member of the subspace and then the last requirement if you remember is that subspaces are closed under multiplication so that if C is a real number and it's just a scalar and if I multiply and v1 is a member of my subspace then if I multiply that arbitrary real number times my member of my subspace v1 I'm gonna get another member of the subspace so it's closed under multiplication these were all of what a subspace is you know there's our definition of the subspace if you call something a subspace these need to be true now let's see if we can do something interesting with what we understand about matrix vector multiplication let's say I have the vector matrix a the matrix a I'm making nice and bold and it's an M by n matrix and I'm interested in the following situation I want to set up the homogenious equation and we'll talk about why it's homogeneous well I'll tell you in a second so let's say we set up the equation my matrix a times vector X is equal to the zero vector is equal to the zero vector this is a homogeneous equation because we have a 0 there homo genius and I want to ask the question I've talked about subspaces is if I take all of the X's if I take the world the universe the set of all of the X's that satisfy this equation do I have a valid subspace so if I take let's think about this I want to take all of the X's that are a member of our and remember if if our matrix a has n columns that I've only defined this matrix vector multiplication if X is a member of our and if X has to have exactly n components only then is it defined so let me define a set of all of the vectors that are a member of RN where they satisfy the equation a times my vector X is equal to the zero vector so my question is is this a subspace is this a valid subspace so the first question is does it contain the zero vector well in order for this to contain the zero vector the zero vector must satisfy this equation so what is any M by n matrix a times the zero vector I'll write it like that times the zero vector well let's write out my matrix a my matrix a a 1 1 a 1 2 all the way to a 1n and then this as we go down a column we go all the way down to a m1 and then as we go all the way to the bottom right we go to a M N and I'm going to multiply that times a unit vector I'm sorry I'm gonna multiply that times the zero vector that has exactly n components so the zero vector with n components is 0 0 and you're gonna have n of these the number of components here has to be exact same number of the number of columns you have but when you take this product this matrix vector product what do you get what do we get well this first term up here is going to be a 1 1 times 0 plus a 1 2 times 0 plus each of these terms times 0 so then you add them all up a 1 1 times 0 plus a 1 2 times 0 all the way to a 1 n times this 0 so you get 0 now this term is going to be a 2 1 times 0 plus a - two times zero plus a23 times zero all the way to a to n times zero that's obviously going to be zero and you're going to keep doing that because all of these are essentially you can kind of view it as the dot product of well if you want if I haven't defined dot product with row vectors in column vectors but I think you get the idea the prot the sum of each of these elements multiplied with the corresponding component in this vector and of course you're just always multiplying by zero and then adding up so you're going to get nothing but a bunch of zeros so the zero vector does satisfy the equation a times the zero vector is equal to the zero vector and this is a very unconventional notation I'm just writing it like that cuz I don't feel like bolding out my zeros all the time to make you realize that that's a vector so we we meet our first requirement the zero vector is a member of this set so it does contain so let me define my set here let me define it n and I'll tell you in a second while I'm calling it n so we now know that the zero vector the zero vector is a member of my set n now let's say I have two vectors V 1 and V 2 that are that set that our members let me write this so let's say I have two vectors V 1 and V 2 and this was a V 2 over here V 1 and V 2 that are both members of our set what does that mean that means that they both satisfy this equation so that means that means that a my matrix a times vector one is equal to zero this is by definition I'm saying that they're member of this set which means they must satisfy this and that also means that a times vector two is equal to our zero vector so in order for this to be closed under addition a times a times vector one plus vector 2 the sum of these two vectors should also be a member of n but let's say figure out what this is the sum of these two vectors is this vector right here this is equal to and I haven't proven this to you yet I haven't made a video where I approve this but it's very easy to prove just using the definition of matrix vector multiplication that matrix vector multiplication does display the distributive property maybe I'll make a video on that but it's literally you just have to go through the mechanics of each of the terms this is equal to a v1 plus a v2 and we know that this is equal to the zero vector and this is equal to the zero vector and if you add the zero vector to itself this whole thing is going to be equal to the zero vector so if v1 is a member of N and v2 is a member of n which means they both satisfy this equation then v1 plus v2 is definitely still a member of n because when I multiply a times that I get the zero vector again so we get let me write that result as well so we are also let me write that so we are also let me write this right here so we now know that v1 plus v2 is also a member of N and the last thing we have to show is that it's closed under multiplication so let's say that v1 is a member of our space that I defined here where they satisfy this equation what about what about C times v1 what about C times v1 is that a member of n well let's think about it what's a our vector our matrix a time's the vector right I'm just multiplying this times a scared I'm gonna get another vector the I don't wanna write a capital V they're lowercase V so it's a vector so what's this equal to well once again I haven't proven it to you yet but it's actually a very straightforward thing to do to show that that the the if when you're dealing with scalars if you have a scalar here it doesn't matter if you multiply the scalar times a vector before multiplying it times the the the matrix or multiplying the matrix times a vector and then doing the scalar so it's very it's straight fairly straightforward to prove that this is equal to C times our matrix a I'll make that nice and bold times our vector V that these two things are equivalent I haven't maybe I should just turn out the video that does this but I'll leave it to you it's very you literally just go through the mechanics by component by component you show this but clearly if this is true if this is true we already know that v1 sorry this was v1 we already know that v1 is a member of our set which means that a times v1 is equal to the zero vector and so that means well this will reduce to C times the zero vector which is still the zero vector so C v1 is definitely a member of n so it's closed under multiplication and you know I kind of assumed this right here but maybe I'll prove it in a different video but I want to do all this to show that this set n is a valid subspace this is a valid subspace that contains the zero vector it's closed under addition it's closed under multiplication and we actually have a special name for this we call this right here we call n we call n the null space the null space the null space of a or we could write n is equal to maybe I shouldn't of written an end written orange in there our orange n is equal to the notation is just the null space of a or we could write the null space is equal the orange notation of N and so literally if I just give you some arbitrary matrix a and I say hey find me and of a what is that you literally you're you are your goal is to find the set of all X's the set of all X's that satisfy the equation a times X is equal to 0 and I'm going to do that in the next video